DRAFT of QUANTUM MECHANIC’S MANUAL

Vladimir Chaloupka, University of Washington December 23, 2019

2

Contents

1 Introduction 7 1.1 The Book Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Instructors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 (Text)books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 General notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Homework, Exams and Grading . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Introduction to Quantum Physics 13

2.1 Experimental evidence for Quantum Physics . . . . . . . . . . . . . . . . . . 13 2.1.1 The humble black body. . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 From Black Body to Blackbody to the Big Bang . . . . . . . . . . . . 15 2.1.3 Max Planck Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.4 blackbody spectrum summarized . . . . . . . . . . . . . . . . . . . . 18 2.1.5 photoelectric effect: . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.6 Compton effect: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.7 wave nature of matter: . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.8 diffraction of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.9 quantization of physical quantities: . . . . . . . . . . . . . . . . . . . 19 2.1.10 atomic spectra: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 The Puzzle Illustrated by the Double Slit Experiment: . . . . . . . . . . . . 20 2.2.1 The Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Fundamentals of Quantum Physics . . . . . . . . . . . . . . . . . . . 21 2.2.3 The Quantum Mechanical Solution of the Puzzle: . . . . . . . . . . . 21 2.2.4 Particle in a potential “box” . . . . . . . . . . . . . . . . . . . . . . 22 2.2.5 Development of quantum physics continued . . . . . . . . . . . . . . 22

3 Quantum Wave Mechanics 23

3.1 Free particle with definite momentum: . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Relations of Uncertainty II: . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.3 Time development of the wave packet for a free particle . . . . . . . . 25 3.1.4 The group velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.5 Particle in a Box revisited . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.6 Elementary Quantum Mechanics and Elementary Relativity: . . . . . 27 3.1.7 calculation of the gravitational redshift: . . . . . . . . . . . . . . . . . 28 4 Quantum Mechanics I 29

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.1 More properties of the wave function . . . . . . . . . . . . . . . . . . 29 4.1.2 Observables ad Operators . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.3 The Eigenequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1.4 Fundamental expression for the QM Probability . . . . . . . . . . . . 30

4.2 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 General properties of the solution . . . . . . . . . . . . . . . . . . . . 32 4.2.3 Consequences for quantization . . . . . . . . . . . . . . . . . . . . . . 32 4.2.4 Current of probability density . . . . . . . . . . . . . . . . . . . . . . 33 4.2.5 (Dis)continuity of the wave-function . . . . . . . . . . . . . . . . . . . 34 4.2.6 Piece-wise constant Potentials . . . . . . . . . . . . . . . . . . . . . . 34

5 Quantum Physics in Hilbert Space 35

5.1 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1.1 Operators and Observables . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1.2 Quantum Physics in Hilbert Space Summarized . . . . . . . . . . . . 36 5.1.3 Hilbert Space Representations . . . . . . . . . . . . . . . . . . . . . . 37 5.1.4 Relations between different representations: . . . . . . . . . . . . . . 39 5.1.5 Expectation value: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.1.6 The Commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.1.7 Structure of the commutator is independent of representation …. 40 5.1.8 General Heisenberg Uncertainty relation . . . . . . . . . . . . . . . . 40 5.1.9 Maximum set of commuting observables . . . . . . . . . . . . . . . . 41 5.1.10 Time-dependence of the expectation values and transition to classical 42 5.1.11 Does Quantum mechanics Satisfy the Equivalence Principle? . . . . . 42 5.1.12 Conservation laws in Quantum Mechanics . . . . . . . . . . . . . . . 43 5.1.13 Conservation of parity from Symmetry of potential . . . . . . . . . . 43 5.1.14 Conservation of Energy from Homogeneity of time . . . . . . . . . . . 44

5.2 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2.1 The ladder operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2.2 The wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2.3 Vector/matrix representation . . . . . . . . . . . . . . . . . . . . . . 46

6 Angular Momentum 49

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.1.1 Commutation properties . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.1.2 The ladder operators, revisited . . . . . . . . . . . . . . . . . . . . . 50 6.1.3 The two families of angular momentum states . . . . . . . . . . . . . 50 6.1.4 Summary of the notation . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.1.5 Addition of angular momenta . . . . . . . . . . . . . . . . . . . . . . 51 6.1.6 Angular Momentum as the Generator of Rotations . . . . . . . . . . 52 6.1.7 Half-integer angular momentum . . . . . . . . . . . . . . . . . . . . . 53 6.1.8 Spin, and Matrix representation . . . . . . . . . . . . . . . . . . . . . 54

6.1.9 Example: Behavior of Spin \frac{1}{2} . . . . . . . . . . . . . . . . . . . . . . 55

7 The atoms 57 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Hydrogen(like) Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2.1 Setting up the H Equation . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2.2 Solving the Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.2.3 The energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.3 The periodic Table of elements . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.3.1 Pauli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.4 Quantum jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.4.1 Atomic physics HW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.4.2 A glimpse at Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . 61 7.4.3 The spin-statistics theorem . . . . . . . . . . . . . . . . . . . . . . . . 61

8 Bell Inequality and Beyond 63

8.1 Entanglement, EPRB and the Bell Inequality . . . . . . . . . . . . . . . . . 63

8.1.1 The Bell Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8.2 Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.3 The Grand Finale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

9 Figures 67

10 Physics and Math Appendix 113

10.1 Math and Physics Survey and Warm-up . . . . . . . . . . . . . . . . . . . . 113 10.1.1 Scientific Notation and a Glimpse at Infinity . . . . . . . . . . . . . . 113 10.1.2 Nicknames for some standard exponents . . . . . . . . . . . . . . . . 115 10.1.3 Note on units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 10.1.4 A glimpse at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 10.1.5 A case study in modern cosmology . . . . . . . . . . . . . . . . . . . 117

10.2 Exponential growth and Exponential decay . . . . . . . . . . . . . . . . . . . 117 10.2.1 Exponential growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.2.2 Exponential decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

10.3 Ranges of selected physical quantities: . . . . . . . . . . . . . . . . . . . . . 118 10.3.1 Mass (in kilograms - kg) . . . . . . . . . . . . . . . . . . . . . . . . . 118 10.3.2 Energy (in Joules - J) . . . . . . . . . . . . . . . . . . . . . . . . . . 119 10.3.3 Time (in seconds - s) . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 10.3.4 Distance (in meters - m) . . . . . . . . . . . . . . . . . . . . . . . . . 120 10.3.5 Scales of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3.6 Atomic structure of matter . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3.7 solar system scale: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3.8 Galactic scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10.3.9 Intergalactic/cosmological scale . . . . . . . . . . . . . . . . . . . . . 121 10.3.10 Scale of densities of matter: . . . . . . . . . . . . . . . . . . . . . . . 122

10.4 Math Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.4.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.4.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.4.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10.5 Complex Numbers: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.5.1 Feynman’s “Amazing Jewel” . . . . . . . . . . . . . . . . . . . . . . . 123 10.5.2 Some applications of the Euler formula . . . . . . . . . . . . . . . . . 124

10.6 The Dirac Delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.6.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10.6.2 The Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

10.7 Mandelbrot Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.7.1 The iterative procedure defining Mandelbrot set . . . . . . . . . . . . 128 10.7.2 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 10.7.3 Complex numbers are very powerful . . . . . . . . . . . . . . . . . . . 130 10.7.4 Mandelbrot, Dirac and Bach . . . . . . . . . . . . . . . . . . . . . . . 133 10.7.5 Vladibrot Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.7.6 Back to the complex Mandelbrot . . . . . . . . . . . . . . . . . . . . 135

10.8 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chapter 1

Introduction

1.1 The Book Notes

Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend things which are there. R.P.Feynman

The point is no longer that quantum mechanics is an extraordinarily (and, for Einstein,

unacceptably) peculiar theory, but that the world is an extraordinarily peculiar place. N.D.Mermin

Quantum Mechanics is one of the most amazing accomplishments of the human spirit. It describes reality, most of which we never see or sense in any direct way, and it shows how the illusion of our everyday life comes about as the ‘classical limit’. We really do not have any right to assume that we should be able to calculate as much as we do, so it is a miracle (a loose quote from Einstein: the most incomprehensible thing about Nature is that it appears comprehensible). Note well that I use the words ‘describe’ and ‘calculate’; how much we really ‘understand’ is a completely different question. We will address this aspect of quantum physics as much as is possible at this stage, but the fact is that you will have to learn a great deal before much speculation on quantum philosophy becomes useful. Some people rejoice in new and strange notions, others get discouraged. If you will find yourself in the latter category, keep in mind that it is worth it: among the countless thousands of experiments on atomic, molecular, condensed matter, nuclear and particle physics (and of course much of chemistry, even astrophysics) there is not a single piece of experimental data in existence which would contradict Quantum Mechanics.

If everything goes well, this Manual - enhanced by your personal notes from lectures and study - should become a comprehensive text for this course. It reflects my ideas about the order of presentation and level of rigour that should be applied at this time. Some topics are mentioned with only a few comments and/or with a summary of key concepts. On the other hand, selected subjects are treated in detail (still, in a non-verbose way). As a rule, it should be possible for you to fill in all the missing steps (and, in general, you will find it very useful to do that !). In addition, the Manual contains a number of Problems; you will also be assigned some Exercises, but you are expected to exercise elementary skills on your own.

Any comments on the Manual, Problems or Exercises (or on anything else) will be much appreciated.

1.2 Instructors

Lecture Instructor: Vladi Chaloupka, Professor of Physics, PAB room B143, vladi@uw.edu, office hours by appointment

Teaching Assistant: TBA

1.3 (Text)books

1) “Quantum Mechanics” by Yoav Peleg, Reuven Pnini and Elyahu Zaarur 2nd edition (joined by Hecht as a co-author, strangely writing the new preface in first person singular ...). This is not your typical textbook: it is part of the Schaum’s Outline series: very compact in the presentation of the theory (that’s why you have to come to class) but very rich in solved examples. It is, by far, the best value for your money (about $15 at Amazon and elsewhere) and it contains an abundance of problems solved in great detail.

2) These Lecture Notes, annotated by you at the lecture and when studying later. Those of you who find the Notes too compact are encouraged to browse the bibliography below, and select a text that you like - much is also available on the web. The three books listed (Griffiths, Bransden/Joachain and Gasiorowicz) were used in this course in the past, so they are recommended. In general, I follow somewhat different sequence from most textbooks - for good reasons, I think. The ability to find a detailed explanation in a book or online when you need it is one of the skills you will need in your studies, as well as in your subsequent career.

Griffiths: Introduction to Quantum Mechanics, 2nd edition. Very popular text, written almost as a transcript of actual lectures. (example: “..... Physically realizable states correspond to the ’square-integrable’ solution to Schroedinger equation. But wait a minute! Suppose I have normalized the wave function ....” ) If you like to be talked to, then this is a book for you. Myself, I cringe at times ....

Bransden/Joachain: Quantum mechanics (2nd ed.) I used this text previously, and covered Chapters 1-6, with some omissions and some additions. Chapter 1 is a very useful recap of Modern Physics missing from the other two books. Very good chapters on various applications of Quantum Physics.

Gasiorowicz: Quantum Physics (3rd ed. Wiley 2003). Used in the past. Now includes many online appendices, but I liked the 1st edition better ... and of course, there is always the (almost) timeless and (almost) perfect classic: Feynman Lectures on Physics!!!

Super-optional readings: Koestler’s Sleepwalkers is the most fascinating book on Science I ever read. The concepts of a scientist as a sleepwalker has been taken over by John Bell of the Bell Inequality fame, and is described beautifully in his ”Speakable and Unspeakable in Quantumn Mechanics.”

Hofstadter’s Godel Escher Bach is, still, the best account of the deep connections between Sciences (including mathematics) and Art. It dies not require any prerequisites, but some pages may take you half an hour or more ... At the end, you will be a better (at least a better educated) person.

And last (and least): a “From Bach to Einstein and Beyond” - a Big Book in progress by yours truly ( faculty XXX). Any comments will be most welcome – this is what I am really concerned about these days ... And to get to know your Instructor up close and personal, you may want to have a look at the first chapter – see two attachments to my email.

1.4 General notes

This course is a core (required) course for participants in the “MS in Applied Physics” program. Some resulting features of the course may be attractive to a wider audience, including students from outside the Physics Department. Since many of our evening students have been out of school for several years, both this course and the EM course (PHYS543) are designed to help re-fresh some possibly forgotten math and physics skills. This means that the subject matter is covered ”from scratch”, and extensive help is available to students who might need it. At the same time, since the audience is more mature and better motivated than typical Junior class, the tempo is brisk, and level is quite rigorous. In short, you can expect to learn a lot (if you are willing to work on it.)

Approach to Math: It has been said: A proof convinces a reasonable man.

A rigorous proof convinces an unreasonable man.

This course will be ”rigorous” in a different way. The proofs which will be given, rather than skipped, will be given not to convince you, but because they are illuminating and/or useful on their own. It will be ”rigorous” not in the sense of splitting hair and enumerating all possible pathological cases when a theorem does not apply (most pathological cases don’t really occur in Nature / Physics), but in the sense that we will try to really understand everything, or clearly recognize what it is that we don’t understand.

1.5 Homework, Exams and Grading

There will be HW assigned in this Manual as well as during lectures. It is VERY IMPOR-TANT to attempt each and every problem BEFORE looking at the solutions (for solved problems in the text, I recommend covering the solutions with a sheet of paper to start with ...).

It is of utmost importance not to get behind. Extensive help is available, but if you wait until a week before Exam, it might be a little late .... HW assignments as sprinkled through the text; some problems are asigned on separate pages. The solutions will be available as indicated on the Course Schedule. The philosophy of this course is

1) ”you will get what you put in”

2) There will be more material presented than can be expected even from a bright and determined student.

3) To serve students with perhaps insufficient preparation, all important topics will be covered from scratch.

I will determine your course grade after I examine

1) your Quantum Diary covering all work you did for the course (PLEASE WRITE LEGIBLY – I cannot grade what I cannot read)

2) your PDF of this Manual personalized by your conclusions, comments, ideas, questions ...

For your annotations please use the Adobe Reader DC or equivalent (enabling text highlighting/underlining, sticky notes, and the identification of the writer (i.e. you).

The grading will be pretty loose, reflecting the course philosophy. In case of any doubt about acceptability of your work please do not hesitate to show me what you are doing as we go along.

Important: When studying the HW problems, recall to use the “strip technique” explained in lecture, to study the solved examples in texts and the Solutions handouts

     PHYS441     Tentative Schedule      Winter2020
Jan    6         Ch. 1,2 Introduction to course and to Quantum Physics
       8
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      13         Ch. 10 delta function, Fourier, complex, vectors/matrices
      15         Ch. 3  Quantum Wave Mechanics
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      20  MLK Day
      22
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      27         Ch. 4  Quantum Mechanics I
      28
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Feb    3
       5          Ch.5  Quantum Mechanics II: Hilbert space
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      10
      12
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      17     Presidents Day
      19
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      24          Ch. 6  Angular Momentum
      26
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Mar    2          Ch. 7  Atoms
       4     reserve
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       9          Ch. 8  Bell Inequality, Quantum Computing,
      11     Grand Finale
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12 CHAPTER 1. INTRODUCTION

Introduction to Quantum Physics

2.1 Experimental evidence for Quantum Physics

The amount of experimental evidence for quantum physics is overwhelming. As noted above, all available data agrees with quantum mechanics, and much of our civilization is actually based on it. In general, I am not a great advocate of a historical approach to science, but there are some foundational experiments of extreme significance and great beauty:

By the end of the 19th century physics has achieved amazing successes:

– Newton, building on contributions of Kepler and Galileo, provided the foundations. unifying everyday mechanics with the celestial motion.

– Maxwell, in a spectacular accomplishment, unified electricity, magnetism and optics. He predicted that a disturbance of the electromagnetic field should travel at a speed of 1/\sqrt{ϵ_{0}μ_{0}} ∼ 3 × 10^{8}m/s. In a truly historic moment, he learned, upon returning from his country house, that astronomers just measured speed of light to be ∼ 3 × 10^{8}m/s.

However, even dramatic and far-reaching developments were awaiting around the corner of the 20th century.

2.1.1 The humble black body.

A black body is an object that does not reflect any light illuminating it – it absorbs everything. But by absorbing the EM energy it gains it and therefore it emits EM energy back. The classical realization has been a cavity with its inner walls maintained at a constant temperature T. After some time an equilibrium is established, where the energy received each second is equal to the energy radiated. The radiation in the cavity can studied by opening a small hole (see Figure 9.2)

It was soon recognized that the equilibrium is characterised by the temperature T the body achieves, and – what is more interesting – the spectrum of that radiation depends only on T and on the frequency f (or equivalentlyλ^{1} on the wavelength

u(f, T) = \frac{c}{f^{2}} u(λ, T)

The independence of the energy spectrum on the blackbody details shows that this is an important, universal aspect of the thermodynamics, and therefore a significant effort centered on find the expression for the energy density u(f, T)

Using very solid classical physics arguments, Rayleigh derived the expression

u(f, T) = \frac{8πf^{2}}{c^{3}} kT  (2.1)

and it was a disaster – see Figure 9.3. It agreed with the experimental data for low frequencies, but it literally exploded, unphysically, at high f. Because no one was able to find fault with Rayleigh reasoning, it became “the ultraviolet catastrophe”.

On the other hand, Wien proposed an expression he found (based on a model that is no longer of interest)

u(f, T) = af^{3}e^{−bf/T}  (2.2)

and this, with the right choice of the adjustable parameters a and b worked quite well (see Figure 9.3). The agreement at high frequency was now perfect (no UV catastrophe!) but now the low frequencies were showing a small but definite and unacceptable discrepancy.

So at the very beginning of the twentieth century there were two formulas; one OK at high frequencies, the other one a low. And in a singular development in physics, Max Planck discovered the correct formula by purely mathematical trial and error:

u(f, T) = \frac{8πh}{c^{3}} \frac{f^{3}}{e^{hf/kT} − 1}  (2.3)

He knew it was correct because it agreed with the experimental data so excellently and over the full range of the variables. In addition, it beautifully made the transition between the Rayleigh and the Wien formulas (see Figure 9.3). So now Planck set out to find out what physics assumptions would yield his result.

After two months of work, he found and – as he was an old-school, careful and conservative^{2} physicist – reluctantly accepted the result: it appeared that the cavity walls emit EM radiation in ”quanta” of energy hf, 2hf, 3hf, ....nhf - and nothing in between. As a consequence, the Boltzman classical average of energy

< E >= ∫ E.P rob(E) = ∫ E \frac{e^{−E/kT}}{∫ e^{−E/kT}}  (2.4)

becomes (with x = hf/kT)

< E >= \frac{∑_{n=0}^{∞} nxe^{−nx}}{∑_{n=0}^{∞} e^{−nx}} = \frac{−x\frac{d}{dx} ∑_{n=0}^{∞} e^{−nx}}{∑_{n=0}^{∞} e^{−nx}} = x\frac{e^{−x}}{1 − e^{−x}} = \frac{x}{e^{x} − 1}  (2.5)

1The dependence on the wavelength seems more familiar; the dependence of frequency makes clear the physics contents of the final Planck solution (see below)

2I say “conservative” in the best sense of the word; more about this in chapter XXX

which, when multiplied by the classical density of states \frac{8πf^{2}}{c^{3}} , becomes the Planck formula. And it is a nice exercise to show it is indeed a smooth transition between Rayleigh and Wien:

lim_{x−>0} eq.2.3 = eq.2.1

lim_{x−>∞} eq.2.3 = eq.2.2

It also is a simple and very nice exercise on integral calculus. Normally, the definite integral is defined as the surface beneath the curve f(x) calculated as a limit of a sum, i.e. the sum is an approximation. But sometimes Nature insists on using the sum, and it is the integral that is an approximation. Equations 2.4 and 2.5, illustrated on Figure 9.5, show how this comes about.

And finally, and remarkably, the Planck formula does not have any free parameter – not even for overall normalization. This was the first crack in the safe where Nature kept the

quantum from us, and Max Planck is immortal for making the breakthrough.

2.1.2 From Black Body to Blackbody to the Big Bang

As fundamental as the blackbody was for the origin of Quantum Physics, there is more – much more.

• Relic from the Big Bang.

As early as 1948, based on not much more that the data of present-day abundances of the light elements (H, D, ...) George Gamow and collaborators predicted that a Big Bang should have left behind a relic of very low temperature radiation. In 1955, in an experiment aimed at reducing the noise interfering with the signals from the first active communication satellites, Penzias and Wilson observed a persistent excess of radiation around the temperature of about 3 K. Fortunately, they knew that Jim Peebles and collaborators were looking for the Big Bang relics in that temperature range. So they persisted, and published their results; Peebles and Co. published the interpretation in the same issue of the journal, and the age of experimental cosmology was born. Penzias and Wilson received a nobel Prize in 1978; Jim Peebles continued a remarkable involvement in cosmology and received an overdue, much deserved Nobel Prize in 2019.

• The COBE spacecraft experiment

Experimental cosmology continued mostly with high altitude balloons, and came of age with the launch of the COBE satellite in 1989. The first great result was obtained very soon: their measurement on the spectrum of the ”relic” - now officially called the Cosmic Microwave Background Radiation (CMBR) produced a fantastic agreement with the Planck formula – see Figure 9.6. By now it was very difficult to think of any other origin of the CMBR than a hot Big Bang. Therefore, this is a good time for a brief overview.

Figure 9.8 shows the (by now) standard chronology of the Big Bang showing how the part of the Universe currently observable by us started extremely small and extremely hot, and in some 13 billions years gradually cooled down and expanded to the present radius of some 42 light years. It is a very busy diagram, and one can (and should) spend a lot of time inspecting and reflecting its many features. It is mind boggling that we can interpolate our knowledge over so many orders of magnitude, and it us only natural that ones degree of confidence decreases as we go closer and closer to the Big Bang itself. Here, we start very conservatively at the end of the period of nucleosynthesis that lasted from about 10^{−10} to 10^{−9} seconds (it sounds really weird to use the word “lasted” in that sentence ...). The Universe was about 13 million light years across and contained a mix of light nuclei, electrons, neutrinos and photons violently agitated at temperature of 10 billion K. And then nothing much happened for about 350 000 years – Universe was gradually expanding and cooling down. And then, quite suddenly, something dramatic happened. As the temperature decreased to about 3,000 K, electrons became for the first time able to combine with the nuclei to produce neutral atoms, and the Universe became transparent. For some obscure reason, this is called “recombination” but those electron and nuclei were never combined previously ... Figures 9.12and 9.9 illustrate the process.

The 3,000 K photons then traveled unimpeded and kept cooling by the continuing expansion until they reached their todays’s temperature T=2.7 K. This is the Cosmic Microwave Background Radiation that Gamow (and others) predicted, Penzias and Wislon (and others) observed, and the COBE experiment could begin to investigate in detail. We already mentioned that their very first result (said to have been achieved in the first nine minutes!) was finding a spectacular agreement of the spectrum with the Planck black body formula. The whole Universe was a black body when CMBR was emitted because it originated in the whole Universe that is an isolated system by definition, and all its constituents were in thermal equilibrium because the expansion rate was much slower than the rate of the thermal motion. In fact, the CMBR is the best black body found in the Universe. Now we will show what else were the experimental cosmologists learn.

• The absolute speed of Earth. By 1992 COBE accumulated enough data for a detailed investigation of the CMBR, Figure 9.7 show that the results were almost too good to be true:

a) the temperature was uniformly 2.7 K in all directions. This was great news for the theory: “we” were originally “in the middle” of everything (because everything was in the middle of everything). Now we are in a ”bubble” of the radius of the observable Universe, and CMBR comes uniformly in as it has been emitted from the inner surface of a sphere called the surface of last scattering (see Figure 9.10).

b) when restricting the data a narrow temperature range of ±3.5mK a dipole signal appeared consistent with the Earth moving through the CMBR background with a speed about 600 km/s. Special relativity is still valid, but clearly there is one reference frame that is more equal than the others :-) c) And finally, the great prize: when removing the dipole signal and restricting the data to ±20μK [sic !!!] there appeared a signal from the radiation of our own galaxy – which is not much more than a nuissance. But the data out of the galactic plane show a clear variation of randomly looking cold and hot spots. This was the Holy Grail, the hope of cosmologists which came to be fulfilled. Of course, the Nobel Prizes did not fail to come (to Mather and Smoot in 2006).

• The (rare) definitive experiment.

The COBE experiment was followed up – with significantly more detailed results – by the WMAP project, and then by a definitive experiment called – appropriately – Planck. The cumulative progress can be seen on Figure 9.11. The pattern of cold and hot spots turned out to be anything but random. In the manner of decomposing a sound signal into contributions of various harmonics, one can similarly express the twodimensional pattern into different “harmonics” that are called “multipoles”. Naturally, the subsequent experiments show more and more details, so we show on Figure 9.13 the results from the Planck experiment. And this is the real Holy Grail. Nature has been kind to us: the peaks of the distribution contain priceless information on various cosmological models and their parameters.

The peaks seen on the Figure are called “acoustic peaks”. It may sound strange that we speak of acoustics in the emptiness of space. But the space was not empty when CMBR was emitted – as you can see from Figure 9.8 the density was considerable, and it was sufficient to enable mechanical waves fully analogous to sound waves in air. But what is important is the amount of information that is provided by the exact locations and heights of the peaks. Ss a dramatic example, it was possible to determine from the spectrum that the metric of the observable Universe if not curved but compatible with flat. This means that the whole Universe is very much larger than our observable Universe, maybe infinite. It is awe inspiring that we can meaningfully discuss the whole Universe that we have never seen and will never see ...

I called the experiment definitive because it cannot be improved. The resolution of the peaks is limited by the physical effects along the path of the CMBR photons from the surface of last scattering to us. When the resolution of the measurements exceeds this limit it will not change the results (unless, of course, the Planck team committed some grave errors, which is very unlikely). So the experiments is definitive and this is rare (and very valuable).

And finally: I find it remarkable to what degree was the theory and the experiment cooperated in achieving the truly breathtaking results that we have discussed. Often it happens that the theory follows the experiment, and the work of the theorists may then have the nature of an after-the-fact postdiction. This significantly reduces the power of any claim that the successful experiments “confirmed” the theory. A dramatic example of the impressive lead of the theory can be seen on Figure 9.14.

In fact, an even much more dramatic example concerns the cosmic inflation. So far we refrained from discussing it because limiting ourselves to the more reliable concepts seemed are-inspiring enough. Inflation is a possibility that the Universe went from initial size of e.g. 10^{−10} to a size of e.g. 10^{+15} meters, within the time of e.g. 10^{−28} seconds; then the “regular” Big Bang followed (well, this would make the Big Bang look not so big, after all -:) This has been motivated by the hope that it would remove some of the remaining difficulties with the theory. It so happens that some (many?) cosmologists now claim that the data from the Black Body research provide experimental evidence for inflation to have in fact happened. As I remarked when discussing the metric of the ”whole Universe”: the mere fact that we can meaningfully discuss the cosmic inflation is mind boggling.

2.1.3 Max Planck Conclusions

Near the end of the Prelude I noted the hope in an eventual recovery of my youthful optimism in our young people, children and grandchildren. Stories like the one I told you in this chapter reinforce that hope. I will close with the words of the Great Man himself: “Science enhances the moral value of life, because it furthers a love of truth and reverencelove of truth displaying itself in the constant endeavor to arrive at a more exact knowledge of the world of mind and matter around us, and reverence, because every advance in knowledge brings us face to face with the mystery of our own being.

2.1.4 blackbody spectrum summarized

The dependence of the ”blackbody spectrum” on wavelength requires quantization of energy levels of electromagnetic radiation inside the black body cavity, with

E = hf h = 6.6 10^{−34} J.s

The ”Planck constant” h = 6.6 × 10^{−34} Js is a fundamental property of Nature, and honors the scientist who discovered it (in addition to the Nobel prize he received). Note the time sequence of Planck’s discovery (correct formula first, its derivation two months later), and do not miss a useful lesson on averaging (see a HW problem in Review of Math).

A fascinating example of black-body radiation is the ”cosmic background” of radiation emitted when, after the Big Bang, radiation ”decoupled” from matter. This happened when Universe was about 400,000 years old, and the temperature was 3000 K. Since then, the expansion of the Universe by a factor of about 1,000 has cooled down the radiation by the same factor to about 3 K today. The precise measurements of the ”ripples” in the cosmic background are starting to provide some incredibly detailed evidence on the early Universe (as well as information on our “absolute” speed).

2.1.5 photoelectric effect:

Electrons are ejected from an anode by a stream of photons, and collected by a cathode. The main results are

1) The collected current depends on the light intensity, but the ”stopping potential” (necessary to prevent any electrons from reaching the cathode ) does not 2) On the other hand, the stopping potential does depend on the frequency of the light used.

eV + W = hf

where the constant h happens to be the same number as in a) as well as below. This might

remind your of Maxwell’s triumph showing the 1/ϵμ = speed of light as measured directly.

Explanation of the photoeffect is what Einstein received his Nobel Prize for. HW: explain to a friend the key non-classical feature of photoelectric effect. You should see, and be able to show to anyone, the extraordinary beauty of the reasoning.

2.1.6 Compton effect:

photon scattering kinematics with

E = hf p = E/c

Recall the general formula

E = \sqrt{m^{2}c^{4} + p^{2}c^{2}}

with the special cases (HW!) of a) rest b) non-relativistic motion (yielding the familiar expression for the kinetic energy) c) ultra-relativistic case (i.e. EM)

2.1.7 wave nature of matter:

2.1.8 diffraction of electrons

they exhibit diffraction as a wave with the wavelength

λ = h/p

Note the serendipitous role of broken equipment, and also the family symbolism:

J.J.Thomson (Nobel prize 1906 for showing quantum nature of electricity)

G.P.Thomson (Nobel prize 1937 for showing wave nature of electrons)

2.1.9 quantization of physical quantities:

2.1.10 atomic spectra:

the spectra of light emitted by excited atoms are quantized as

hf_{ab} = E_{b} − E_{a}

where, as discovered later, E_{b} and E − a are the energies of the initial and final state (i.e. there was a ”quantum jump” from a to b). We will understand that soon enough.

Stern-Gerlach experiment:

the deflection of particles going through inhomogeneous magnetic field are quantized as if the had the angular momentum projection quantized as:

L_{z} = mh/2π = mℏ

2.2 The Puzzle Illustrated by the Double Slit Experiment:

(Feynman Vol. III,1.-3.) see our Fig. 2

To fully appreciate the Puzzle, you should try to ‘comprehend’ that if you open the slit B, electrons will stop arriving at point P !

HW: impress your friends with this, the Central Mystery of Quantum Physics, using a modified version of the double-slit experiment using the “Mach-Zehnder” interferometer. This not only illustrates the puzzle directly and quantitatively, but it also serves as an example of the central role that interferometry (looking at a tiny differences plays in physics.

2.2.1 The Interferometer

Light impinges on a half-silvered mirror BS (for ”Beam Splitter” !). The rest of the spectrometer consists of two ordinary mirrors M1 and M2, detectors D, D1 and D2 and a screen, as indicated on Fig. . We experimentally observe to following:

1) If D1 and D2 are as on Figure, each detector will detect light of one half of the intensity recorded by detector D if inserted. So far so good: the mirror BS split the light in beams going along Path1 and Path2.

2) If we diminish the incident flux below a certain level, it turns out that the light comes in some sort of quanta, and D1 and D2 never fire simultaneously. Well, we already know that quanta pf lights are phonons, so that mirror BS simply transmitx some of them, and reflects the others (that’s why it is called ”half-silvered”)

3) If we remove D2, D1 will keep firing as before, and screen will get hit by equal (within statistics) flux, and the impacts will be distributed in a broad smooth peak as illustrated on Figure. You say: of course, the photons that went along path2 now making all the way to the screen. Same result if we keep D2 and remove D1.

4) And now comes the surprize: if we remove both D1 and D2, the distribution of hits on the screen will show dark and bright fringes. In particular, there are spots on the screen that were getting hit when D1 was in and D2 out stop to be getting hit when we just remove D2.

5) When in case 4) we do something to path 1 and not to path 2 (the simplest is to introduce a gradual delay along path 1) the whole spectrum gradually shifts.

The results in 4) and 5) show a big problem: we cannot say that some photons came along path1 and others along path2. Because of 2) we cannot say that each photon has split and went along both paths. SOMETHING seems to have gone along BOTH path1 and path2, but it was not “the” photon. We obtain similar results using other particles (neutrons, atoms, even molecules). Note that this is not some weird quantum theory: these experimental facts will have to be explain by the present or ANY future theory (recall the quotes on top of the first page). Using particles traveling slower than light, we can have fun of thinking about the ontology when we place. The path one detector to the position D′_{1}. Whenever this detector does not detect the particle when predicted, we know that (and exactly when) will the detector D_{2} hit. And last but not least: So what happened? As we shall see, our present theory (quantum mechanics) claims that this question is meaningless. The formalism allows us to correctly calculate the probabilities of the outcome of any experiment we can perform, the experiments discussed above are different experiments, and ”what really happened” is, at best, a question for philosophers (while the ”real physicists” say: “Shut up and calculate” :-).

2.2.2 Fundamentals of Quantum Physics

Summary of experiments

1) wave-particle duality

                      bullets      water waves       electrons      EM waves
lumps(quanta) ?          yes           no               yes        YES
interference ?            no           yes              YES        yes

(the answers in upper case are the essence of the quantum-mechanical particle-wave duality)

2) Many physical quantities are ”quantized”.

2.2.3 The Quantum Mechanical Solution of the Puzzle:

Planck (1901) : E = hf to explain blackbody spectrum

Einstein (1905) : p = E/c = hf/c = h/λ to explain photoelectric effect

deBroglie (1925) : a (complex) wave with λ = h/p also for electrons etc. (all of the above with the same value of the Planck constant h =)

Born (1927) : probabilistic interpretation of ψ

momentum p, energy E ⇐⇒ complex ψ(x, t) with wavelength λ = h/p and period T = h/E

We shall see how boundary conditions impose quantization. A bit later, we shall see how all this is embedded in the abstract, elegant theory of Quantu Physics in Hilbert Space.

2.2.4 Particle in a potential “box”

Already at this stage we can do some quite fundamental physics. Consider a very simple but very useful ”playground” of Quantum Mechanics (see Fig. 8). Particle of mass m is confined in a potential V (x) = 0 for 0 < x < a and V (x) = ∞ outside. Standing waves with

L = nλ_{n}/2

immediately yield quantization of energy

E_{n} = n^{2}E_{1}

where E_{1} = h^{2}/8mL^{2} ≠ 0

This is our first encounter with quantization as consequence of the boundary conditions. The hydrogen atom will turn out to be just like this (only a little more complicated mathematically). The remarkable, highly non-classical result that the ground energy is not zero will turn out to be required by the Uncertainty Principle.

2.2.5 Development of quantum physics continued

Born rule elaborated

It would be hard to overestimate the importance of the Born rule giving the correct interpretation of the wave function. It is interesting to note that Born, in his Nobel lecture, gave credit to Albert Einstein.

| ψ(x) |^{2} dx ≡ probability of finding the particle in (x, x + dx)

∫_{a}^{b} | ψ(x) |^{2} dx ≡ probability of finding the particle in (a, b)

| ψ(x) |^{2}≡ probability density

ψ(x) ≡ probability amplitude or the “possibility wave”

for a discrete case, |ψ_{i}|^{2} is the probability of the i-th outcome

several distinguishable alternatives: add probabilities

for several indistinguishable alternatives: add probability amplitudes i.e. the “possibility” waves or “wave functions”

NOTE: When you think about it, the alternatives to get a given (i.e. fully specified) quantum state from a given (fully specified) initial state are ALWAYS indistinguishable. Failure to appreciate this is at the origin of many published papers.

Quantum Wave Mechanics

3.1 Free particle with definite momentum:

traveling waves: to satisfy both the Planck/Einstein condition E = hf and the deBroglie relation λ = h/p, we construct a plane wave:

ψ(x, t) = Ae^{i(kx−ωt)} = Ae^{i(px−Et)/ℏ} = Ae^{i(x/λ−t/T)2π}

with p = ℏk, k = 2π/λ, and E = E(p) (ω = ω(k)). ℏ = h/2π (or Physics dept. coffee bar.) Complex numbers appreciation here, and elsewhere (see Review of Math).

Wave Packets to describe a Localized Particle:

physics: localisation in the x-space vs. localisation in the p-space (k-space)

math: the probability amplitude F(k) and the probability amplitude f(x) are related by the Fourier transform:

f(x) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{+∞} F (k)e^{ikx}dk F (k) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{+∞} f(x)e^{−ikx}dx

where F(k) is the ”cooking recipe” specifying the amount (and phase) of the momentum ℏk in the wave packet.

Relation of uncertainty I:

from counting number of wiggles in a finite region, and estimating the uncertainty in the count to be at least 1/10 of a wiggle, we get

δ_{x}δ_{p} ≥ h/10

(see Fig. 7) (doing the same in the (t,f) domain, we obtain the ’acoustical Heisenberg’:

δ_{t}δ_{f} ≥ 1/10

3.1.1 Relations of Uncertainty II:

math: for any two variables x, k related by a Fourier transform it is possible to show ^{1} that

δ_{x}δ_{k} ≥ 1/2

. notes:

It follows that

δ_{x}δ_{p} ≥ ℏ/2

This is not far off from the Uncertainty I !

It also follows that

δ_{E}δ_{t} ≥ ℏ/2

but interpretation needs some care. Also, there is an interesting connection to line widths and lifetimes. And in acoustics, we get

δ_{t}δ_{f} ≥ 1/4π

As we shall see, interpretation of all uncertainty relations needs a lot of care.

Sometimes δq = ∞ . Even in this case, a restriction on the uncertainty of k can be found 2H.J.Landau and H.O.Pollak, Bell Syst. Tech.J. 40 (1961) 65.

Deep down, the uncertainty relations result from non-commuting operators: more about this (and also about the original Heisenberg argument about disturbing the system!) in the section on Uncertainty Relations III. All this is just a tip of the iceberg (vacuum fluctuations,

...).

3.1.2 Examples

Example 1: a flat rectangular packet

f(x) = \sqrt{\frac{1}{L}} in (−\frac{L}{2}, \frac{L}{2}) ⇒ F (k) = \sqrt{\frac{2}{πL}} \frac{sin(kL/2)}{k}

The packet is centered in momentum space on k=0. Therefore, it does not move —it just diffuses. The packet peaks at k = 0, but it has many “lobes” all the way to ±∞. This is the consequence of the sharp edges of the packet in x-space. It is amusing to see the 0/0 behaviour at k = 0. It is also amusing to check the uncertainty relation. Note: obviously, a flat rectangular packet in k-space will produce a similar peak with lobes in the x-space. However, the time dependence will be different in the two cases.

Example 2: a chopped beam

f(x) = \sqrt{\frac{1}{L}}e^{ik_{0}x} in (−\frac{L}{2}, \frac{L}{2}) ⇒ F (k) = \sqrt{\frac{2}{πL}} \frac{sin((k − k_{0})L/2)}{(k − k_{0})}

1see E.H.Kennard,Z.Phys.44 (1927) 326

Same as Example 1, but the packet is now moving with momentum ℏk_{0}.

Example 3: the Gaussian wave packet ^{3}recall: a Gaussian distribution with mean y_{0} and standard deviation σ is

\frac{1}{σ\sqrt{2π}} e^{−(y−y_{0})^{2}/2σ^{2}}

The meaning of σ is: “the half-width at 61% of maximum”.

F (k) = \sqrt{\frac{2a}{\sqrt{2π}}} e^{−a^{2}(k−k_{0})^{2}} ⇒ f(x) = \frac{1}{\sqrt{a\sqrt{2π}}} e^{ik_{o}x}e^{−x^{2}/4a^{2}}

Interpretation: p = ℏk_{0} , σ_{x} = a , σ_{k} = 1/2a ⇒ σ_{x}σ_{p} = 1/2 , i.e. σ_{x}σ_{p} = ℏ/2. The product of the two uncertainties is the minimum consistent with the Relation of Uncertainty.

It is very useful to visualize the wave packet as a little “screw” with x along Ref(x), y along Imf(x) and the particle moving along z. Thanks to the complex numbers, f(x) is wavy as required by deBroglie, but does not ever go through zero (as required by the probability interpretation of a free particle. Note also the the width of the wave packet has nothing to do with the size of the particle!

3.1.3 Time development of the wave packet for a free particle

The relations that explain the deep connections between wavelength, period, momentum, energy, wave number and angular frequency are so important that we repeat them here::

f(x, t) = Ae^{2πi(x/λ−t/T)} = Ae^{i(kx−ωt)} = Ae^{2πi(px−Et)}

The phase velocity:

For a plane wave: f(x, t) = e^{i(kx−ωt)} = e^{ik(x−v_{phase}t)}, where v_{phase} = ω/k.

3.1.4 The group velocity

At any time t, the maximum of the wave packet

f(x, t) = \frac{1}{\sqrt{2π}} ∫ F (k)e^{i(kx−ω(k)t)}dk (note that for free particle F (k, t) = F (k))

will be at a point where

\frac{d(kx − ω(k)t)}{dk} = 0 ⇒ x = \frac{dω}{dk} t ⇒ v_{group} = \frac{dω}{dk}

3) Specific result Important example of Gaussian wave packet

for a Gaussian packet and ω(k) = ω_{0} + v(k − k_{0}) + β(k − k_{0})^{2} (these are the first three terms of a Taylor expansion, i.e.

v = dω/dk|_{0} β = 1/2 d^{2}ω/dk^{2}|_{0}

A little (in fact, a moderate amount of) algebra results into:

f(x, t) = e^{i(k_{0}x−ω_{0}t)}\sqrt{\frac{π}{a^{2} + iβt}}e^{−(x−vt)^{2}/4(a^{2}+iβt)}

This is a particle moving with momentum p_{0} = ℏk_{0}, with probability density corresponding to a Gaussian with mean = x − vt and standard deviation

σ = a\sqrt{1 + t^{2}/τ^{2}} where τ = a^{2}/β

notes:

for a relativistic particle (E = pc), you get v = c and no spreading (β = 0). Note that our β is not the usual β = v/c.

for a non-relativistic particle (e = p^{2}/2m), you get v = p/m and τ = 2ma^{2}/ℏ. This is a very long time for even a very small macroscopic particle. Notice that it is proportional to the square of the original position uncertainty.

The expression for σ can be written as

σ = \sqrt{σ_{0}^{2} + σ_{1}^{2}}

where σ_{0} = a is the original uncertainty in position at time t = 0, and σ_{1} = βt/a = tσ_{p}/m = σ_{v}t is the uncertainty due to the uncertainty in the momentum. Note that σ = σ_{0} initially, and σ = σ_{0} much later - just like at the Boston marathon!

3.1.5 Particle in a Box revisited

This is a very simple but very useful ”playground” of Quantum Mechanics (see Fig. 8). Standing waves with

L = nλ_{n}/2

immediately yield quantization of energy

E_{n} = n^{2}E_{1}

where E_{1} = h^{2}/8mL^{2} ≠ 0, in agreement with the relations of uncertainty

HW: Use the Uncertainty Relation to estimate the energy of the ground state of the Simple Harmonic Oscillator. (hint: use the Uncertainty Relation to express < p^{2} > in terms of < x^{2} >, and then look for a minimum of E).

3.1.6 Elementary Quantum Mechanics and Elementary Relativity:

calculation of Doppler shift using a radiative decay:

This example and associated homework provide a wealth of quantum physics, relativity and numerical methods for fun and profit. Consider a decay (see Fig. 9) of a particle with (mass,momentum,energy)=(m_{1}, p⃗_{1}, E_{1}) into a particle with (m_{2}, p⃗_{2}, E_{2}) and a photon with (0, p⃗, E). Recall the Lorentz factors

β_{1} = \frac{v_{1}}{c} = \frac{p_{1}c}{E_{1}}

γ_{1} = \frac{1}{\sqrt{1 − β_{1}^{2}}} = \frac{E_{1}}{m_{1}c^{2}}

Now, the energy and momentum conservation give

E_{1} − E = E_{2}
c(p⃗_{1} − p⃗) = cp⃗_{2}

When you subtract the square of the second equation from the square of the first one, you should get (after recalling that |a⃗ + b⃗|^{2} = a^{2} + b^{2} + 2abcosθ)

E = \frac{m_{1}^{2}c^{4} − m_{2}^{2}c^{4}}{2m_{1}c^{2}} \frac{1}{γ_{1}(1 − β_{1}cosθ)}

Physical interpretation of this result becomes clear if we re-write it as

E = E_{0} \frac{\sqrt{1 − β_{1}^{2}}}{1 − β_{1}cosθ}

where E_{0} is the energy of the photon when the parent particle is at rest, i.e. it is the energy of the photon as seen from the rest-frame of the parent particle. And now comes the gem: the relation between the photon frequency f as seen from the laboratory frame, and the frequency f_{0} as seen from the rest frame of the parent particle is obtained by using

E = hf, E_{0} = hf_{0}

which yields the Doppler formula

f = f_{0} \frac{\sqrt{1 − β_{1}^{2}}}{1 − β_{1}cosθ}

HW: fill in the missing steps.

3.1.7 calculation of the gravitational redshift:

This is a much simpler but equally remarkable connection of elementary quantum physics to relativity (this time, General Relativity). Consider light of frequency f_{0} emitted downwards from a tower of height x. At the bottom, each photon will have energy E = E_{0} + gxE_{0}/c^{2}. This immediately yields the gravitational redshift

(f − f_{0})/f_{0} = gx/c^{2}

Note that this does not depend on h. On Earth, with x = 20m, gx/c^{2} = 2.10^{−15}. This has been measured(!) using the Mössbauer effect.

Quantum Mechanics I

4.1 Introduction

To progress towards a more detailed quantum mechanics, we have two questions to deal with:

a) what are the properties of the state described by a particular function ψ(x)

b) what is the ”equation of motion” for ψ(x, t)

4.1.1 More properties of the wave function

In quantum physics, the wave function ψ(x, t) describes the system completely. This means we can use it to determine not just the probability density P (x, t) = |ψ(x, t)|^{2} but also the possible values a_{i} of any physical observable A and their probabilities. This is done by a fairly involved and completely new procedure in three steps:

4.1.2 Observables ad Operators

to every observable A we assign an operator Â that operates on a wave function and produces another function:

Âψ(x, t) = ϕ(x, t)

Examples: for position we have x̂ = x, the momentum is p̂ = −iℏ∂/∂x For now, you should take this as an “experimental” fact; we will see the deep logic motivating these choices later in this course.

4.1.3 The Eigenequation

The only possible values that observable A can have are solutions of the equation

Âψ_{a}(x, t) = aψ_{a}(x, t)

This equation is called an eigen-equation, and its solutions are the eigenvalue (a) and corresponding eigenfunction (ψ_{a}).

Example 1: for the function ψ_{1}(x) = sin(px/ℏ) we have

p̂ψ_{1}(x) = −iℏ\frac{d sin(px)/ℏ)}{dx} = −ip cos(px/ℏ)

Therefore, ψ_{1} is not an eigenfunction of momentum. Similarly for ψ_{2}(x) = cos(px/ℏ)

Example 2: for the function ψ_{3}(x) = e^{+ipx/ℏ} we get

p̂ψ_{3} = pψ_{3}

and ψ_{3} is an eigenfunction of momentum, with eigenvalue p

similarly, for the function ψ_{4}(x) = e^{−ipx/ℏ} we get

p̂ψ_{4} = −pψ_{4}

and ψ_{4} is an eigenfunction of momentum, with eigenvalue −p

Example 3: HW: check that all four functions ψ_{1}(x) - ψ_{4}(x) are eigenfunctions of kinetic energy, all with the same eigenvalue E = p^{2}/2m. This is our first encounter with the important concept of “degeneracy”, where more than one eigenfunction correspond to the same eigenvalue.

Example 4: As HW, formulate and solve the eigen-equation for the position operator x̂. Note that this is very simple, but also very subtle (and very important)!

4.1.4 Fundamental expression for the QM Probability

If the system is in the state described by ψ and we measure the observable A, the probability amplitude for obtaining the particular value (a) is

∫_{−∞}^{∞} ψ_{a}^{∗}(x)ψ(x)dx

Immediately after finding the value “a”, the state of the system will change from ψ to ψ_{a}. This is the (in)famous “collapse” of the wave function. Much more on this (and on the whole “measurement problem”) later in this course.

4.2 The wave equation

The equation describing the time development of the wave function between measurements is the celebrated Schroedinger equation ^{1}:

iℏ\frac{∂ψ(x, t)}{∂t} = −\frac{ℏ^{2}}{2m} \frac{∂^{2}ψ(x, t)}{∂x^{2}} + V (x, t)ψ(x, t)

1Remarkably, Schrodinger’s interpretation of his own equation was wrong, and reaction by his competitor Heisenberg and others was not enthusiastic, to say the least. But he had the last laugh ...

This may look very complicated, but once you recall that the operator of momentum is p̂ = −iℏ∂/∂x you can see right away that the right hand side is simply

(\frac{p̂^{2}}{2m} + V̂)ψ = Êψ = Ĥψ

In words: the rate of change of the wave function with time is given by the operator of kinetic plus potential energy, i.e. operator of the total energy Ê, also known as the Hamiltonian Ĥ, operating on the wave function.

Note that in contrast to Newton’s law, the Schrodinger equation is first-order in time, so that ψ(x, t_{0}) suffices to specify ψ(x, t).

Simple as it may be, the equation is so difficult to solve that an analytic solution is known only for a handful of cases; the rest of the problems must be solved using all kinds of approximations, or by a brute force (numerically).

4.2.1 Separation of variables

A very important method, both theoretically as well as in practice, of reducing the complexity of the problem is available for time-independent potentials: V (x, t) = V (x). By using a beautiful trick we will encounter several times (also in EM) one can easily show that there are particular solutions of the Schroedinger equation that have the form

Ψ(x, t) = ψ(x)ϕ(t)

where ψ and ϕ are solutions of the ordinary differential equations

\frac{dϕ(t)}{dt} = −i\frac{E}{ℏ} t ϕ(t)

and

Êψ(x) = Eψ(x)

where E is an arbitrary (for now) constant The first equation has a ready solution, the same for any problem, namely:

ϕ(t) = e^{−iEt/ℏ}

The second equation must be solved for the particular potential V (x) that appears in the Hamiltonian. Note that it really is an eigen-equation for energy, also called Time-Independent Schroedinger Equation, or TISE, as opposed to TDSE. So the “arbitrary” constant is really the energy E, and the equation should be written in the form

Ĥψ_{E}(x) = Êψ_{E}(x) = (\frac{p̂^{2}}{2m} + V̂)ψ = Eψ_{E}(x)

where ψ_{E}(x) is the energy eigenfunction corresponding to energy E.

The highlight of all this is that since the equation is linear, a general solution can then be assembled of the particular solutions:

Ψ(x, t) = ∫_{E} or ∑_{E} g(E)ψ_{E}(x)e^{−iEt/ℏ}

with the “weighting function” g(E) chosen to reproduce the initial conditions of the given problem. Note that another name for g(E) is therefore the “probability amplitude” for energy, |g(E)|^{2} is the probability density, and |g(E_{n})|^{2} would be the probability of energy E_{n}.

4.2.2 General properties of the solution

Before we discuss specific solutions, it will be useful to pause for a moment to think about their general properties. Recast the TISE as

\frac{d^{2}ψ(x)}{dx^{2}} ∼ −(E − V)ψ(x)

When particle moves in the classically-allowed region, i.e. E > V, the equation implies that the second derivative of ψ has opposite sign to that of ψ itself. A minute with a pen and pencil will show you that the behavior will be oscillator, with agreement with Dr.(also Prince) deBroglie. The wavelength of the oscillations will vary with the value of (E-V) but as long as it is positive, ψ will oscillate.

On the other hand, the region with E < V is classically forbidden - the kinetic energy is negative, so momentum is imaginary! But the SE does have a non-trivial solution, with second derivative of the same sign as ψ. And again, you should be able to convince yourself that in this case it implies an exponentially-suppressed penetration of the particle into the forbidden region. And this is a far-reaching conclusion, obtained quite early in our discussions!

4.2.3 Consequences for quantization

But there is more we can already deduce and learn. Imagine a particle in a potential well as illustrated on Fig., together with the qualitative behavior of its wave function we now understand. We will easily prove (next section) that at the points where particle crosses between allowed and forbidden regions, the wave function and its first derivative must be continuous (as long as the potential there is finite). It turns out that this is only possible at selected values of the energy: E_{1}, E_{2}, E_{3}, .... This is the physical origin of quantization of the atomic levels, stability of the atom, and much more!

On the other hand, if the particle is not restricted from both sides, the conditions are achievable for any energy, and particle is ”unbound” - free to move, but not “really” free if V (x) ≠ const.

Piece-wise constant potentials

a) The simplest case of this important category is when there is only one piece: V (x) = V_{0}. This means the particle is free, and we have already obtained the wave function on p. 9 - convince yourself that it indeed satisfy the Schroedinger equation (as well as the demands of Prof. Einstein on the period, and Prof. deBroglie on the wavelength). The wave packet we discussed (also on p. 9) as an example of the Fourier Transform, is now seen as a special case of the assembly of the general solution from the particular solutions obtained by the separation of variables.

b) The second simplest case: V (x) = V_{0} on an interval, and V (x) = ∞ anywhere outside the interval is nothing but the ”particle-in-the-box” we discussed on p. 11. It is educational to think how this is modified when the box is not infinitely deep.

c) The general piece-wise-constant potential is also the most useful: it contains reflection off a potential wall, scattering off a potential well, and - most importantly - tunneling through a potential barriere. In any of these cases, the solutions are analytically known in any region of the constant potential: where E > V, the wave function is

ψ_{E}(x) ∼ e^{±ikx} where k^{2}ℏ^{2}/2m = E − V

and for the forbidden region

ψ_{E}(x) ∼ e^{±kx} where k^{2}ℏ^{2}/2m = V − E

Assembling the overall wave function from the individual segments, as illustrated on Fig. for the case of tunneling, and demanding the continuity of ψ and its derivative dψ/dx at the entry and exit of the barriere yields four (4) algebraic equations for the constants B,C,D and E (why don’t we need to know A, and why is there not F?). And finally the sweet reward: all this is for for the stationary case of one particular solution. Now assemble a real, moving wave packet, and sit back and watch what ”really happens” (DEMO). The variety of effects is amazing, and covers much of modern quantum electronics!

Note also that many potentials can be approximated in a piece-wise constant manner. In principle, this can get you an easy approximate solution of the corresponding problem (the handling of the system of algebraic equations can be made easy by the matrix technique as shown in our Texts).

4.2.4 Current of probability density

Multiply the Schrodinger equation by ψ^{∗} and subtract the complex-conjugate of this construct. Show (as HW) that this yields

∂ρ(x, t)/∂t = −∂j(x, t)/∂x

where ρ(x, t) = ψψ^{∗} is the familiar probability density, and

j(x, t) = (ℏ/m) Im[ψ^{∗}(x, t)∂ψ(x, t)/∂x]

Integrating the equation over x from a to b gives

dP_{ab}(t) = j(a, t) − j(b, t)

which shows that the interpretation of the function j(x, t) is the “current (flux) of probability density”. The differential and the integrated equations are the differential and integral expression of the conservation of probability. In particular, for a particular choice (which one?) of a and b, this implies conservation of normalization.

HW: Using the polar form of the wave function: ψ(x, t) = a(x, t)e^{iϕ(x,t)} show that:

a) the result for the current of the probability density of a free particle makes perfect sense.

b) to obtain non-zero current of probability density, the wave function must be ”non-trivially complex”.

4.2.5 (Dis)continuity of the wave-function

1) The wave function itself must be continuous everywhere (otherwise the current of probability density would have a singularity). We express this as

disc_{a}ψ(x) = lim_{ϵ→0}[ψ(a + ϵ) − ψ(a − ϵ)] = 0

2) The discontinuity of the first derivative is obtained from the Schrodinger equation:

\frac{∂ψ}{∂x} |_{a+ϵ} − \frac{∂ψ}{∂x} |_{a−ϵ} = ∫_{a−ϵ}^{a+ϵ} dx\frac{∂^{2}ψ}{∂x^{2}} = \frac{2m}{ℏ^{2}} ∫_{a−ϵ}^{a+ϵ} dx[V (x) − E]ψ(x)

Therefore the derivative is continuous at x = a for a finite potential V (a), and for a singular potential V (x) = λδ(x − a) we get

disc_{a} \frac{∂ψ}{∂x} = \frac{2mλ}{ℏ^{2}}ψ(a)

(the parameter λ represents the “strength of the singularity”.)

4.2.6 Piece-wise constant Potentials

Quantum Physics in Hilbert Space

5.1 Hilbert Space

It so happens (this should be taken as an “experimental” fact) that the space of quantummechanical states is a Hilbert space: a vector space consisting of abstract vectors |ψ > , |ϕ >, ... with “reasonable properties” for addition and multiplication of vectors by complex numbers (i.e. commutative, distributive and associative laws, plus existence of a unique null vector). In addition, a bilinear “inner product” (a generalization of the ordinary scalar product of two vectors) is defined, with

< ϕ|ψ >=< ψ|ϕ >^{∗}

As we shall see, the “dual” vectors ^{1} are generalizations of complex conjugation.

5.1.1 Operators and Observables

Operators in Hilbert space operate on vectors to produce different vectors:

A|ψ >= |ϕ >

An operator can operate on a bra or on a ket, according to the definition

< ϕ|(A|ψ >) ≡ (< ϕ|A)|ψ >≡< ϕ|A|ψ >

The definition of operator A^{†} Hermitian-adjoint to operator A is ^{2}

< ϕ|A|ψ >=< ψ|A^{†}|ϕ >^{∗}

1Since the form < | > looks like a bracket, Dirac applied his “physicist’s sense of humor” and called the vector < | a bra and the vector | > a ket. The space of the bra’s is said to be “dual” to the space of the kets. The power of Dirac notation comes from the ability to put the full definition of the state inside the | >, so you may have |ψ > or |ground state of the Hydrogen atom > etc. A numerical, quantitative representation of the abstract Hilbert state vector |ψ > may then be a complex function ψ(x) =< x|ψ > etc.

2Unfortunately, many texts confuse this simple but subtle issue: Ohanian (an otherwise excellent text) writes < α|Q|β >= (< α|Q^{†})|β > which is technically incorrect. Griffiths (eq. 3.15 - 3.17), Gasiorowicz (eq. 6-9) and many others write < Aϕ|ψ >=< ϕ|A^{†}|ψ > which is confusing at best: in Dirac notation, ψ is NOT a vector in Hilbert space - it is just a label, and an operator operating on a label does not make sense.

and if A = A^{†} we say that the operator A is Hermitian. It is easy to see that the expectation values and the eigenvalues of a Hermitian operator are real (do it as HW). Therefore, physical observables are always represented by Hermitian operators.

It is easy to see that the eigenvectors of a Hermitian operator corresponding to different eigenvalues are mutually orthogonal (the case of degeneracy is discussed in the section on the Commutator):

A|ψ_{1} >= a_{1}|ψ_{1} >
< ψ_{2}|A =< ψ_{2}|a_{2}
< ψ_{2}|A|ψ_{1} >= a_{1} < ψ_{2}|ψ_{1} >
< ψ_{2}|A|ψ_{1} >= a_{2} < ψ_{2}|ψ_{1} >
⇒ 0 = (a_{1} − a_{2}) < ψ_{2}|ψ_{1} >

Examples:

• The set of all square-integrable functions is a Hilbert space, with the inner product

defined as

< f|g >= ∫ f^{∗}(x)g(x)dx

example of an operator would be p = −iℏ\frac{∂}{∂x}

HW: prove that the operator p is Hermitian (and appreciate, again, the importance of complex numbers in quantum physics).

• The set of n-dimensional complex vectors is a Hilbert space with the inner product

< a|b >= ∑_{i=1}^{n} a_{i}^{∗}b_{i}

In this case, operators will have the form of an n × n matrix.

HW: show that an operator A is hermitian if A_{ij} = A_{ji}^{∗}.

We will see how these (and other) examples appear as representations of the abstract Hilbert space of quantum-mechanical states.

5.1.2 Quantum Physics in Hilbert Space Summarized

Here is an (informal) summary of the formal postulates of Quantum Mechanics: ^{3}

I. States of physical systems are represented by vectors in Hilbert space. Any linear combination (with complex coefficients) of possible states is a possible state. This is the Principle of Superposition (which is due to the linearity of the Schrodinger equation), and it leads to most of the quantum-mechanical conundrums.

3In the following, we omit the “hat” above the operator symbol when there is no danger of confusion, so instead of Â we will simply use A.

II. The time development of a state is given by the Schrodinger equation

iℏ\frac{d|ψ >}{dt} = H|ψ >

where H is the Hamiltonian (operator of energy) of the system. Note that - in contrast to the Newton’s law - this is a first-order equation. Note also that time t appears, but position x does not - “existence happens in the Hilbert space”. In other words, the non-relativistic quantum mechanics is non-local from the very start.

III. The probability that if we prepare the system in state |ψ > we will find it in state |ϕ > is

P = | < ϕ|ψ > |^{2}

where < ϕ|ψ > is the “probability amplitude”.

IV. Any observable A is associated with a Hermitian operator A, and all possible outcomes of a measurement of A are the eigenvalues of A. After the measurement, the system will be in the corresponding eigenstate:

A|ψ_{a} >= a|ψ_{a} >

Va. The set of eigenvectors of a Hermitian operator is complete: an arbitrary state |ϕ > can be expressed as a linear combination (a sum or an integral - see Section 3a and 3b below):

|ϕ >= ∑_{a} c_{a}|ψ_{a} >

Vb. The set of eigenvectors is (or can be made) orthonormal:

< m|n >= δ_{mn} < q|q′ >= δ(q − q′)

for discrete or continuous spectrum, respectively.

5.1.3 Hilbert Space Representations

When we choose a “basis”, i.e. a complete set of orthonormal states (a discrete set of |m >, or a continuous “set” of |x >), we obtain a particular representation of the abstract Hilbert space.

a) discrete case: the orthonormality is expressed by < m|n >= δ_{mn}. Completeness means that any state |ψ > can be expressed as a linear combination of the base states |n >

|ψ >= ∑_{n} c_{n}|n >

where the complex coefficients c_{n} can be easily determined by making inner product of this equation with < m|, and using the orthonormality:

< m|ψ >=< m| ∑_{n} c_{n}|n >= ∑ c_{n}δ_{mn} = c_{m}

This means that the expansion can be written as

|ψ >= ∑_{n} < n|ψ > |n >= (∑_{n} |n >< n|)|ψ >

Therefore, the operator

I ≡ ∑ |n >< n|

is an identity operator, and can be inserted in expressions for the inner product or for the expectation value, in the eigen-equation etc. Consequently, in the discrete case, states are represented by state vectors

c_{m} =< m|ψ >

while operators are represented by matrices ^{4}

A_{mn} =< m|A|n >

and eigen-equations reduce to matrix equations

∑_{n} A_{mn}c_{n} = ac_{m}

This is the “matrix” (Heisenberg) formulation of Quantum Mechanics (compared to “wave” (Schrödinger) formulation. It will be very useful for description of spin.

b) In the continuous case, the orthonormality condition is < q|q′ >= δ(q −q′). Completeness means that any state |ψ > can be expressed as a linear combination of the base states |q >:

|ψ >= ∫ dq c(q)|q >

where the complex coefficient c(q′) can be determined by making inner product of this equation with < q′| and using the orthonormality:

< q′|ψ >=< q′| ∫ dq c(q)|q >= ∫ dq c(q) < q′|q >= ∫ dq c(q)δ(q − q′) = c(q′)

C_{ij} = ∑_{k} A_{ik}B_{kj}

4Recall that the first index (i) in a matrix element A_{ij} represents the row, and the second index (j)

represents the column. So for example, a multiplication of two matrices C = AB is verbalized (do it!) as follows: “the element at the intersection of i-th row and j-th column is the scalar product of i-th row of the first matrix with the j-th column of the second matrix”.

This means that the expansion can be written as

|ψ >= ∫ dq < q|ψ > |q >= (∫ dq|q >< q|)|ψ >

Therefore, the operator

I ≡ ∫ dq|q >< q|

is an identity operator, and can be inserted in inner products, expectation values, eigenequations etc. States are represented by state functions

ψ(x, t) =< x|ψ >=< ψ|x >^{∗}

or ψ(p, t) =< p|ψ > etc., i.e. any given state |ψ > can have many equivalent “representations” < x|ψ >, < p|ψ >,...). Operators are represented by (uncountably infinite-

dimensional)“matrices” A(x, x′) =< x|A|x′ >. It is easy to se that

< x′|x̂|x′′ >= x′δ(x′ − x′′)

so that x̂ = x. Example:

< x >= ∫ ψ^{∗}(x)xψ(x)dx

It is somewhat more difficult (we will skip the proof) to show that, in the x-representation, the momentum operator is p̂ = −iℏ∂/∂x.

5.1.4 Relations between different representations:

The transformations between different representations are accomplished by using the resolution of the identity again. Notice how the familiar notions such as coordinate system rotation, or the Fourier transform, are all just a special case of the Hilbert space representation change:

ψ(x) =< x|ψ >= ∫ dp < x|p >< p|ψ >= ∫ dp e^{ipx/ℏ}ϕ(p)

ψ(x) =< x|ψ >= ∑_{n} < x|n >< n|ψ >= ∑_{n} ξ_{n}^{∗}(x)ψ_{n}

ψ_{n} =< n|ψ >= ∫ dx < n|x >< x|ψ >= ∫ dx ξ_{n}^{∗}(x)ψ(x)

ψ_{n} =< n|ψ >= ∑_{m′} < n|m′ >< m′|ψ >= ∑_{m′} c_{nm′}ψ_{m′}

(note: the meaning of the terms in the last column can be seen from the middle column: the first example is a transformation from a continuous base to a different continuous base (in this case specifically p → x), second example is discrete → continuous, third example is continuous → discrete, and last is discrete → discrete. In the last example, n labels one set of the base states {n}, and m′ labels another set of base states {m′}.

5.1.5 Expectation value:

Show, as HW, that the expectation values are

< A >_{ψ} = < ψ|A|ψ > = ∑ a_{i}|c_{i}|^{2} = ∫ dq a(q)|c(q)|^{2}

where a_{i} is the eigenvalue corresponding to the i-th eigenstate of A in the discrete case, a(q) is the eigenvalue corresponding to the “q-th” eigenstate of A in the continuous case, and c_{i} or c(q) are the corresponding probability amplitudes.

5.1.6 The Commutator

The commutator of two operators

[A, B] ≡ AB − BA

plays a fundamental role in Quantum Physics. The following four subsections explain this role in some detail:

5.1.7 Structure of the commutator is independent of representation

It is easy to see that the structure of the commutation relation is the same in any representation: if in the Hilbert space [Â, B̂] = Ĉ then in any representation [A, B] = C. In particular, if [Â, B̂] = c where c = complex number (i.e. the operator c ∗ Î), then [A, B] = c in any representation (recall the example of [x̂, p̂] in the x- and p-representations).

5.1.8 General Heisenberg Uncertainty relation

Define the uncertainty of A by

δA = \sqrt{< (a− < a >)^{2}} = \sqrt{< A^{2} > − < A >^{2}}

Then define operators U ≡ A− < A > and V ≡ B− < B > and convince yourself that

δA = \sqrt{< U^{2} >}  δB = \sqrt{< V^{2} >}  [U, V] = [A, B]

Now investigate, as HW, the behavior of the quantity

I(λ) ≡< U + iλV |U + iλV >

for real values of λ. Obviously, I(λ) ≥ 0 for any value of λ. Show that I(λ) is smallest (but of course still non-negative) for

λ = −i\frac{(< [U, V] >)^{2}}{2 < V^{2} >}

(and pause for a moment to show that this indeed is a real number for any two Hermitian operators U and V.) Then plug it into the definition of I(λ) and show that the grand result is: For any two Hermitian operators A and B, in any state | ψ > :

δA.δB ≥ \frac{1}{2} |< ψ | [A, B] | ψ >|

a) if A and B do not commute, this gives the most general form of the uncertainty relation (this is of course the famous Heisenberg Uncertainty Relation). Note the precise meaning: given an ensemble of many identical copies of a quantum system, you measure A on some of them, and B on the others. The resulting distributions of measured values a and b obey the Heisenberg Inequality. Note that this is a different interpretation than that of professor Heisenberg himself (he considered performing both measurements on the same event, and the first measurement disturbed the system ...).

b) if A and B do commute, this expression says that the corresponding observables can be measured simultaneously. This corresponds to the fact that commuting operators have common eigen-functions (see next section).

5.1.9 Maximum set of commuting observables

Commuting operators have common eigen-functions: Apply B to both sides of A|ψ >= a|ψ >, and commute A with B to obtain

AB|ψ >= aB|ψ >

This means that the state |ϕ >= B|ψ > is an eigenstate of A, with eigenvalue a.

a) the non-degenerate case: there is only one independent eigenstate of A corresponding to eigenvalue a. Therefore, |ϕ > must be proportional to |ψ >, i.e. |ϕ >= B|ψ >= b|ψ >, i.e. |ψ > is also an eigenstate of B.

b) the degenerate case: consider two independent eigenstates of A corresponding to the same eigenvalue. Dropping the hats and brackets to save some typing, we have:

Aψ_{1} = aψ_{1} and Aψ_{2} = aψ_{2}

As in case a), the states Bψ_{1} and Bψ_{2} are eigenstates of A, and can therefore be expressed as some linear combinations of the independent eigenstates ψ_{1} and ψ_{2} of A:

Bψ_{1} = α_{1}ψ_{1} + β_{1}ψ_{2} and Bψ_{2} = α_{2}ψ_{1} + β_{2}ψ_{2}

Is there ϕ = ψ_{1} + λψ_{2} such that Bϕ = bϕ ? By equating the coefficients multiplying ψ_{1} and ψ_{2}, you should obtain a quadratic equation for λ. There are therefore two solutions to the eigenequation for B, each with different eigenvalue b (but the same eigenvalue a of A). The two eigenstates ϕ are therefore orthogonal. In the general case, this procedure is repeated until all degeneracy is removed. A maximum set of mutually commuting operators is called a complete set, and its eigenvalues fully specify a quantum-mechanical state. (Simplest example: the two-fold degeneracy of the energy of free particle, removed by specifying of eigenvalue of momentum (do as HW).)

5.1.10 Time-dependence of the expectation values and transition to classical

The Commutator governs the time-dependence of the expectation values: It is easy to show (HW) from the Schrödinger equation that

\frac{d < A >}{dt} =< \frac{∂A}{∂t} > +\frac{i}{ℏ} < [H, A] >

In particular, this means that if an observable, not explicitly dependent on time, commutes with the Hamiltonian, then its expectation value is conserved, in any state. This is different from the notion of stationary state (an eigenstate of the Hamiltonian), where the expectation value is conserved for any observable.

Schrödinger equation and Newton’s law

Using the above equation it is easy to show (HW) that

\frac{d < x >}{dt} = \frac{< p >}{m} and \frac{d < p >}{dt} = − < \frac{dV}{dx} > = < F (x) >

This is very similar to the Newton’s law dp/dt = F but we are not quite there yet. For the true correspondence we need < F (x) >= F (< x >) and this is the subject of the Ehrenfest principle:

For potentials that are slowly varying over the region where the wave-function is appreciable:

− < \frac{dV (x)}{dx} >= −\frac{dV (< x >)}{d < x >} = F (< x >)

proof:

< F (x) > = < F (< x >) + (x− < x >)F′(< x >) + \frac{1}{2}(x− < x >)^{2}F′′(< x >) + ... > ≅

≅ < F (< x >)+ < x− < x >> F′(< x >) = F (< x >)

and we get

m\frac{d^{2} < x >}{dt^{2}} = F (< x >)

in words: the center of a quantum wave-packet moves according to the Newton’s law (even in purely quantum-mechanical regime !). Note that the restriction to slowly varying potentials is essential: recall the reflection from a potential step!

5.1.11 Does Quantum mechanics Satisfy the Equivalence Principle?

Note that the Schrödinger equation for a particle in gravitational field

iℏ\frac{∂ψ}{∂t} = −\frac{ℏ^{2}}{2m} \frac{∂^{2}ψ}{∂x^{2}} + mgψ

contains an explicit mass dependence which does not cancel out. Although the motion of the center of mass is independent of m:

\frac{d^{2} < x >}{dt^{2}} = g

one might expect a mass dependence to turn up in a sufficiently subtle experiment. This is indeed the case: consider the setup where a neutron can go from A to D either via B or via C (see Figure). It is easy to calculate the phase difference Δϕ = ϕ_{ABD} − ϕ_{ACD}, as a function of the tilt-angle δ. Since the phase difference on segments CA and BD is equal, it is sufficient to compare segments AB and CD. From energy conservation

E = p^{2}/2m = p^{′2}/2m + mgl_{2}sinδ

one gets

Δϕ = l_{1}Δk ≅ l_{1}(k^{′2} − k^{2})/2k = −m^{2}gl_{1}l_{2}λsinδ/2πℏ^{2}

The same result can be obtained from Δϕ = ωΔt . Remarkably, this effect has indeed been observed using a neutron interferometer.

5.1.12 Conservation laws in Quantum Mechanics

Recall that if an operator A, not explicitly dependent on time, commutes with the Hamiltonian, its expectation values are conserved. In particular, if

Aψ(x, 0) = aψ(x, 0) then Aψ(x, t) = aψ(x, t).

This can be nicely illustrated with the help of the formal solution of the Schrödinger equation with a time-independent Hamiltonian:

|ψ(t >= e^{−iHt/ℏ}|ψ(0) >

This gives:

< ψ(t)|A|ψ(t) > = < ψ(0)|e^{iHt/ℏ}Ae^{−iHt/ℏ}|ψ(0) > = < ψ(0)|A|ψ(0) > .

5.1.13 Conservation of parity from Symmetry of potential

The implication of an operator commuting with the Hamiltonian on the structure of the Hamiltonian is best explained on the example of the parity operator:

0 = [H, P]ψ(x) = [V (x), P]ψ(x) = V (x)P ψ(x) − P V (x)ψ(x) = V (x)ψ(−x) − V (−x)ψ(−x)

Therefore, for parity to be conserved, the potential must be symmetric (V (x) = V (−x)).

Conservation of momentum from Homogeneity of space

In general, each symmetry (invariance in respect to a transformation) has a “generator” which corresponds to a conserved observable. As an important example, consider the translational invariance (i.e. homogeneity of space). The generator of a translation in space is the momentum operator: infinitesimal translation:

ψ(x + ϵ) = ψ(x) + ϵ\frac{df}{dx} = T (ϵ)ψ

where T (ϵ) = 1 + ϵ\frac{d}{dx} = 1 + \frac{i}{ℏ}pϵ

finite translations:

ψ(x + a) = T (a)ψ(x) where T (a) = lim_{n→∞} (1 + \frac{i}{ℏ} \frac{a}{n}p)^{n} = e^{ipa/ℏ}

For Schrödinger equation to be invariant in respect to translations:

T (iℏ\frac{∂ψ}{∂t}) = T (Hψ) ⇒ iℏ\frac{∂(T ψ)}{∂t} = H(T ψ)

T̂ must commute with Ĥ, i.e. p̂ must commute with Ĥ, i.e. momentum will be conserved.

5.1.14 Conservation of Energy from Homogeneity of time

You can now see that the meaning of the Schrödinger equation is: “generator of the translations in time is the Hamiltonian.” This is explicitly seen comparing the expression for T with the formal solution of the Schrödinger equation

ψ(x, t) = U(t)ψ(x, 0) where U(t) = e^{−itH/h}

Homogeneity of time thus leads to the conservation of energy. Similarly, invariance in respect to rotations (i.e. isotropy of space) will lead us to the concept of angular momentum and to its conservation.

5.2 Simple Harmonic Oscillator

This section is of great practical as well as theoretical and pedagogical importance. The SHO potential

V (x) ∼ x^{2} + p^{2}

occurs in a wide variety of physics situations, either directly, or as an approximation close to a local minimum. The familiarity with the Hilbert space method of the SHO solution is an absolute necessity for student to master dealing with commutation relations, and it is a great precursor to a very similar treatment of the angular mpmentum. Therefore the whole section is assigned as HW: students will be expected to fill in the missing steps and missing constants (such as m, ω, ℏ etc.) and a problem testing your understanding may well come in the 2nd midterm or in the Final Exam. Everything is treated in great detail in Griffiths section 2.3 - 2.3.1, as well as in Pnini section 5.4 and Solved Problems 5.8-5.10.

5.2.1 The ladder operators

The procedure starts with switching from operators x and p to the new operators of the form

5

a_{±} ∼ x ∓ ip

These operators do not represent any physical quantity as they are not Hermitian (a_{+}^{†} = a_{−} ≠ a_{+}). Instead, we use them, and the inverse relationship

x ∼ (a_{−} + a_{+}) p ∼ (a_{−} − a_{+})

to obtain the expressions for the commutator and for the Hamiltonian:

[a_{−}, a_{+}] = 1 H = ℏω(a_{±}a_{∓} ± \frac{1}{2})

Assuming that |E > is an eigenfunction of the Hamiltonian with the eigenvalue E, it is then easy and in fact extremely elegant to show that

H(a_{+}|E >) = ℏω(a_{+}a_{−}a_{+} + \frac{1}{2} a_{+})E >= ℏωa + (a_{+}a_{−} + 1 + \frac{1}{2})|E >= a_{+}(H + ℏω)|E >=

= (E + ℏω)(|a_{+}|E >)

This means that the state a_{+}E > is an eigenstate of H with eigenvalue increased by the amount ℏω (=hf !): a_{+}|E >∼ |E + ℏω > Similarly, the operator a_{−} decreases E to E − ℏω. We then we apply the lowering operator a_{−} repeatedly, until we reach the ground state that cannot be lowered, i.e.

a_{−}|E_{0} >= 0

and the actual value of the ground state energy is immediately obtained from

H|E_{0} >= ℏω(a_{+}a_{−} + \frac{1}{2} >= \frac{1}{2} ℏω|E_{0} >

So there you have it: we have obtained (in Hilbert space !) the spectrum of SHO as equidistant energy levels with spacing hf sitting on the ”zero-point energy” E_{0} = \frac{1}{2}hf > (due, in final analysis, to the Heisenberg Uncertainty Principle). And then all higher sates are obtained by repeated application of the raising operator a_{+} on |E_{0} >.

Going from Hilbert space to the coordinate representation, the equation a_{−}|0 > becomes a simple differential equation

a_{−}ψ_{0}(x) ∼ (x + ip)ψ_{0}(x) = 0

with the immediate solution

ψ_{0}(x) ∼ e^{−x^{2}}

5Note that Pnini (and many other texts) denote Griffiths’ operators a_{−} and a_{+} by a and a^{†}, respectively

It can be verified by direct integration of |ψ_{1}|^{2} (see Griffiths Example 2.4) that the normalization coefficient for ψ_{1} is 1.0, i.e. in Hilbert space, changing notation from |E_{n} > to |n >

a_{+}|0 >= |1 >

The normalization for higher states is then obtained from realizing that

a_{+}a_{−}|n >= n|n > a_{−}a_{+}|n >= (n + 1)|n >

so that if < n|n >= 1 and < n + 1|n + 1 >= c then

< n|a_{−}a_{+}|n >=< n|(a_{−}a_{+}|n >) = (n + 1) = (< n|a_{−})(a_{+}|n >) = c^{2}

so that

a_{+}|n >= \sqrt{n + 1}|n + 1 > a_{−}|n >= \sqrt{n}|n − 1 >

It is fun to watch how the repeated application of the raising operator then produces all the eigenstates

|n >= \frac{1}{\sqrt{n!}}(a_{+})^{n}|0 > corresponding to E_{n} = (n + \frac{1}{2})ℏω

5.2.2 The wave functions

Going back to the coordinate space: all eigenstates ψ_{n}(x) inherit the Gaussian exponential factor from ψ_{0}(x), corresponding to the penetration into the left and right forbidden regions, multiplied by an n-th degree polynomial obtained by the repeated addition ox x and differentiation; physically this corresponds to the ”wiggles” required by Dr. deBroglie as the particle moves in the allowed region between the turning points. The polynomials are important in many physics applications, and are called the “Hermite polynomials”. Figure ?? shows several lowest wavefunctions, as well as a high-n wavefunction to illustrate the transition to the classical regime. Note that even the large-n state will not oscillate - this needs an addition of at least two states of different energy. Note also that a wave packet consisting of many states (e.g. n=91, 92, ....109, 100) will oscillate at the frequency corresponding to the energy difference of two consecutive states!

5.2.3 Vector/matrix representation

Finally, it is interesting to change from the coordinate representation to the energy representation , where states are represented by the (“ordinary”) vectors ^{6}:

|0 >→ (1, 0, 0, 0, ....)
|1 >→ (0, 1, 0, 0, ....)
|2 >→ (0, 0, 1, 0, ....)

6I am typesetting these a row-vectors, but they really should be vertical column vectors ...

5.2. SIMPLE HARMONIC OSCILLATOR 47 and so on, and operators are represented by matrices, such as

x̂ → x_{nm} =< n|x|m >∼< n|(a_{+} + a_{−})|m >

Such representation will be crucial when we discuss the states with half-integer angular momentum.

48 CHAPTER 5. QUANTUM PHYSICS IN HILBERT SPACE

Angular Momentum

6.1 Introduction

The operator of angular momentum is a straightforward generalization of the classical expression:

L̂ = r̂ × p̂

Its components, using the Levi Civita symbol ^{1} are

L̂_{i} = ∑_{j} ∑_{k} ϵ_{ijk}r̂_{j} p̂_{k}

(HW: use this to write explicit expressions for the three components)

We will also need the operator for the square of the magnitude of the angular momentum. Omitting the hats, this is

L^{2} = L_{x}^{2} + L_{y}^{2} + L_{z}^{2}

We will have to use the above definition expressed in spherical coordinates ^{2} (Griff. Fig. 4.1 and eqs. 4.123 - 4.132). Note how much more complex these equations (and the expression for the Laplacian - Griff. eq. 4.13) are in spherical coordinates than in the Cartesian coordinates. The exception is eq. 4.129:

L_{z} = −iℏ\frac{∂}{∂ϕ}

HW: Derive this equation, and ponder its connection to the earlier one dimensional definition

p = −iℏ\frac{∂}{∂x}

1defined as 1) ϵ_{123} = +1; same for the two cyclic permutations of indices (ϵ_{312} = ϵ_{231} = +1) 2) ϵ_{ijk} changes sign on an exchange of two indices It follows that the “anti-cyclic permutations ϵ_{321} = ϵ_{132} = ϵ_{213} = −1 and all other ϵ_{ijk} = 0 (i.e. those with at least one pair of equal indices). The full power of this symbol will become apparent in your E&M course.

2Note that x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ and dΩ = r^{2} sin θ dr dθ dϕ

6.1.1 Commutation properties

It is easy to show that the three components of the operator L̂do not commute with each other, but each of them does commute with L^{2}:

[L_{i}, L_{j}] = iℏϵ_{ijk} L_{k} [L^{2}, L_{i}] = 0

Therefore, we can choose one component (traditionally L_{z}) and find the common eigenfunctions with L^{2}:

L^{2}|λ, μ >= λ|λ, μ >
L_{z}|λ, μ >= μ|λ, μ >

6.1.2 The ladder operators, revisited

This proceeds, remarkably enough, along the lines of Simple Harmonic Oscillator. Defining L_{±} = L_{x} ± iL_{y} we get

L_{z}(L_{+}|λ, μ >= (μ + ℏ)L_{+}|λ, μ >

This means that the state L_{+}|λ, μ > corresponds, up to a normalization constant, to the state |λ, μ+ℏ >. But this raising of μ must stop when we reach a certain value μ_{max}, because a projection of a quantity cannot exceed the quantity itself, and therefore

L_{+}|λ, μ_{max} >= 0

This, and subsequent use of L_{−}, leads to the explicit construction of all states. Changing the labeling of the states from |λ, μ > to |l, m >, the final result is

L^{2}|l, m >= l(l + 1)ℏ^{2}|l, m >
L_{z}|l, m >= mℏ|l, m >

where −l ≤ m ≤ l with unit distance between the consecutive values of m.

6.1.3 The two families of angular momentum states

An obvious solution is a family of integer values on m: l = 0, m = 0 l = 1, m = -1,0,1 l = 2, m = -2,-1,0,1,2 etc. As in Simple Harmonic Oscillator, again, the explicit representations by functions Y_{l}^{m}(θ, ϕ) are determined from L_{+}Y_{l}^{l} = 0, and Y_{l}^{m−1} ∼ L_{−}Y_{l}^{m}. These so called “spherical harmonics” are easy to obtain if you are patient - they get complicated quite quickly - see Griffiths Table 4.3, and Figures.

The second, less obvious family of solutions is for half-integer values of l:

l = \frac{1}{2}, m = −\frac{1}{2}, \frac{1}{2} l = \frac{3}{2}, m = −\frac{3}{2}, −\frac{1}{2}, \frac{1}{2}, \frac{3}{2} etc

Remarkably, these states cannot be due to any orbital motion ^{3} . So the family of solutions of the commutation relations is larger than the operator r̂ × p̂ that was at the origin of the commutators. And even more remarkably, these states indeed exists in Nature - in fact you are made of them (protons, neutrons and electrons all have “inherent angular momentum” \frac{1}{2})!

6.1.4 Summary of the notation

Traditionally, we distinguish three kinds of angular momentum by using symbols L, S and J:

L^{2}, L_{z} (with eigenvalues labeled by l, m) denote “orbital angular momentum”. The values of l are restricted to integers 0,1,2,... S^{2}, S_{z} (with eigenvalues labeled by s, m) denote “spin” - the angular momentum in the rest-frame of the particle J^{2}, J_{z} (with eigenvalues labeled by j, m) denote generic angular momentum.

Spin (or generic angular momentum) may be integer or half-integer (of orbital nature, of inherent (quantum) nature, or hybrid).

Most importantly, several identical particles of integer spin (“bosons”) obey the “Bose-Einstein statistics”: the wave function is symmetrical upon any exchange. On the other hand, several identical particles of half-integer spin (“fermions”) obey the “Fermi-Dirac statistics”: the wave function is anti-symmetric upon any exchange, i.e. changes sign. Therefore, two (or more) identical fermions cannot be in the same quantum state. As we shall see, this (the “Pauli exclusion principle”) is the foundation of the periodic table, and in fact foundation of all atomic physics.

6.1.5 Addition of angular momenta

We will need to be able to deal with several angular momenta in the same system. Classically, this is just a sum of the angular momentum vectors of the individual components (such as for example Moon’s spin around its axis, combined with its orbital momentum as it goes around the Earth). As you might expect, in QM it gets a little more tricky: addition of angular states |j_{1}, m_{1} > and |j_{2}, m_{2} > results in one of the states |j, m > with m = m_{1} + m_{2} and |j_{1} − j_{2}| ≤ j ≤ (j_{1} + j_{2}). The full treatment is quite tricky, but fortunately in this class we will only need to analyze the simplest case of j_{1} = j_{2} = \frac{1}{2}. This enables us to use a very simple notation: we will need the relation between:

the representation using |j, m > (four states: |1, 1 >, |1, −1 >, |1, 0 > and |0, 0 >) and the different but equivalent representation using |m_{1}, m_{2} > (four states: |\frac{1}{2}, \frac{1}{2} >, | − \frac{1}{2}, \frac{1}{2} > , |\frac{1}{2}, −\frac{1}{2} > and | − \frac{1}{2}, −\frac{1}{2} >)

3An elegant way to see that is to show that L_{−}Y \frac{1}{2} \frac{1}{2} ≠ Y − \frac{1}{2}_{\frac{1}{2}} . HW: do this, or at least think about how would you do it.

The first two states are easy: clearly |1, 1 >= |\frac{1}{2}, \frac{1}{2} > and |1, −1 >= | − \frac{1}{2}, −\frac{1}{2} > The state |1, 0 > can be obtained by applying the lowering operator J_{−} on the state |1, 1 >. Using the action of J_{−} derived above^{4}, we obtain

|1, 0 >= \frac{|\frac{1}{2}, −\frac{1}{2} > +| − \frac{1}{2}, \frac{1}{2} >}{\sqrt{2}}

The last state is

|0, 0 >= a|\frac{1}{2}, −\frac{1}{2} > +b| − \frac{1}{2}, \frac{1}{2} >

and the values of the coefficients a and b are obtained from orthonormality:

< 1, 0|0, 0 >= 0 = (a + b)\sqrt{2} < 0, 0|0, 0 >= 1 = a^{2} + b^{2}

yielding

|0, 0 >= \frac{|\frac{1}{2}, −\frac{1}{2} > −| − \frac{1}{2}, \frac{1}{2} >}{\sqrt{2}}

The remarkable properties of this state will lead us, in the Grand Finale, to the famous Bell Inequality.

6.1.6 Angular Momentum as the Generator of Rotations

Upon an infinitesimal rotation of the coordinate system around the z-axis by angle ϵ, a function f transforms as

f′(x, y, z) = f(x − ϵy, y + ϵx, z)

= f(x, y, z) − ϵy \frac{∂f}{∂x} + ϵx\frac{∂f}{∂y} = (1 + iϵL_{z}/ℏ)f(x, y, z) = D_{z}(ϵ)f(x, y, z)

A finite rotation is then obtained by the operator

D_{z}(ϕ) = lim_{n→∞}(D_{z}(ϕ/n))^{n} = e^{iL_{z}ϕ/ℏ}

⇒ D_{z}(ϕ) |jm⟩ = e^{imϕ} |jm⟩

This equation will be your main tool in the homework ....

Notes

- The above results use the Taylor expansion of f(x − ϵ, y + ϵ, z) around (x, y, z), and the binomial expansion of an exponential -The similarity of the operator of translation in the z-direction

D_{z}(a) = e^{ip_{z}a/ℏ}

4Note that when applying J_{−} on the right-hand side, you must apply it only the first particle, then only on the second particle, and add the results.

is not accidental: recall that in spherical coordinates L_{z} = −iℏ∂/∂ϕ. -We see that the operator L_{z} is a ’generator of rotations around the z-axis’. This turns out to be independent of the particular representation, i.e. it is true in the Hilbert space. It is also valid for situations in which angular momentum cannot be described in a coordinate representation at all, i.e. for ‘spin’. -To get the ’active rotation’ (i.e. rotation of the object), change ϕ to −ϕ. -The physics consequences: if H is invarient with respect to rotations around z ⇒ [H, D_{z}] = 0 ⇒ [H, L_{z}] = 0 ⇒ L_{z} is conserved. This is the same reason which leads to the conservation of p̂ if H is invariant with respect to translations. -The same results are obviously obtained for the x- and y-projections. An arbitrary rotation is obtained by a suitable combination of rotations around around different axes, and an invariance of the Hamiltonian with respect to such a rotation leads to the conservation of the angular momentum Ĵ. -Now you can see that two rotations do not, in general, commute because their generators do not commute, which is due to the fact that [L_{z}, L_{y}] ≠ 0. (In classical physics, rotations are represented by matrices, and we know matrices in general do not commute).

- Consider two coordinate systems with axes z and z′. Obviously, either |jm⟩ or |jm′⟩ form complete sets of orthonormal states. Therefore

|jm′⟩ = ∑_{m} |jm⟩ ⟨jm|jm′⟩ = ∑_{m} c_{mm′} |jm⟩

and

|jm⟩ = ∑_{m′} |jm′⟩ ⟨jm′|jm⟩ = ∑_{m}^{′} c_{m′m} |jm′⟩

where the coefficients c_{ij} have the usual probability interpretation. Note that the above transformation between two bases in Hilbert space is analogous to the relation between two mutually rotated bases in an ordinary vector space. Recall another example of such a transformation – the Fourier transform. Note also that it is easy (and instructive) to transform the matrix representing an operator to the new basis.

6.1.7 Half-integer angular momentum

The angular momentum operator Ĵ cannot be represented as r̂× p̂ in the case of half-integer j. The see this, look at the action of raising and lowering operators J_{±} = J_{x} ± iJ_{y} in the spherical coordinate representation in which

|jm⟩ → Y_{jm}(θ, ϕ)

and

J_{±} → ℏe^{±iϕ}(±\frac{∂}{∂θ} + i cot θ\frac{∂}{∂ϕ})

The equations

J_{±}Y_{j±j} = 0

and

J_{z}Y_{j±j} = ±jℏY_{j±j}

can easily be shown to have the solution

Y_{j±j}(θ, ϕ) ∼ e^{±ijϕ} sin^{j} θ

You should verify that Ĵ^{2}Y_{j±j} = j(j +1)ℏ^{2}Y_{j±j}. More importantly, you should verify that for integer j, you can consistently go up and down the ladder, and thus obtain all the spherical harmonics for given value of j. However, for j = \frac{1}{2}:

J_{+}Y_{\frac{1}{2} − \frac{1}{2}} ∼ e^{iϕ}(\frac{∂}{∂θ} ± cot θ\frac{∂}{∂ϕ})\sqrt{sin θ}e^{−iϕ/2} ∼ \frac{cos θ}{\sqrt{sin θ}}e^{iϕ/2} ≠ Y_{\frac{1}{2}} _{\frac{1}{2}}

The same contradiction occurs for any half-integer j. Therefore, states with half-integer j cannot be due to the orbital angular momentum. Another way to see this is from the approach terminating the Legendre polynomial, (again analogous to a similiar approach to the Harmonic Oscillator), which was seen to yield states with integer j only–see Text). Miraculously enough, states with half-integer angular momentum do exist (protons, neutrons, electrons, quarks,...), and what they are due to is a good question. It is easy to see, but not so easy to comprehend. that the j = 0 states cannot be the only fundamental building blocks, but the j = \frac{1}{2} states might be.

6.1.8 Spin, and Matrix representation

We have seen that only the integer angular momentum states can be represented by state functions of coordinates, which are then acted on by differential operators. However, all angular momentum states can be represented by bracketing the state with the eigenstates of the angular momentum. A state |ψ⟩ of angular momentum j is then represented by a (2j + 1)-dimensional vector

ψ_{n} = ⟨jn|ψ⟩

and operators are represented by (2j + 1) × (2j + 1) matrices

A_{mn} = ⟨jm|A|jn⟩

The matrix elements of Ĵ^{2} and Ĵ_{z} are trivial, the matix elements of J_{±}^{†}J_{±} provide the correct normalization, and matrix elements of J_{±} serve to determine the matrix elements of J_{x} and J_{y}. We will define spin as angular momentum of a particle in its own rest frame. Particles with integer spin are called bosons, particles with half-integer spin are called fermions. Spin angular momentum of a fermion cannot be completely ’due’ to the orbital angular momentum of its components. A particle with spin can of course move in space. It will then be described by a vector (spinor) with elements ψ(r⃗, t) which will obey the time-dependent Schroedinger equation

iℏ\frac{∂ |ψ⟩}{∂t} = Ĥ |ψ⟩

where the Hamiltonian operator will be a (2j + 1) × (2j + 1) matrix.

6.1.9 Example: Behavior of Spin \frac{1}{2}

This example is mathematically the easiest case. To compensate for this, the physics interpretation is more difficult, as it demonstrates the fundamentally non-classical nature of half-integer spin.

First we need to establish the correct normalization of the spin states. By a procedure along the normalization of the states of SHO we obtain

S_{±}|s, m >= ℏ\sqrt{s(s + 1) − m(m ± 1)}|s, m ± 1 >

Then we use this to get the matrix elements of the operators S_{x}, S_{y}, S_{z} and S^{2} - see Griffiths p. 174 Then we use the general procedure for solving a matrix eigen-equation for an N by N operator B:

B|ϕ >= b|ϕ >

This is equivalent to a system of N homogeneous equation for the N unknown coefficients d_{m}

∑_{m} d_{m}(b_{mn} − bδ_{mn}) = 0 n = 1, N

This system of equations has a non-trivial solution if

det|b_{mn} − bδ_{mn}| = 0

This is a polynomial equation for b of order N, with N solutions. Corresponding to each solution there is a set of coefficients d_{i} corresponding to an eigenfunction |ϕ_{i} > of B.

For s = \frac{1}{2} the procedure is mercifully simple, and results (as expected) into eigenvalues of

m_{z} = ±\frac{1}{2}, m_{x} = ±\frac{1}{2}, m_{y} = ±\frac{1}{2}

corresponding to eigenvectors

[(0, 1), (1, 0)], [\frac{1}{\sqrt{2}}(1, ±1)], [\frac{1}{\sqrt{2}}(1, ±i)]

Note the three different ways in which orthogonality is achieved. Next obtain the expression for the eigenfunction |m_{x} = +\frac{1}{2}⟩ in the coordinate frame rotated around the z-axis by ϕ:

|ψ′⟩ = \frac{e^{iϕ/2}}{\sqrt{2}} (1, e^{−iϕ})

and use it to answer the following questions:

a) Determine ⟨S_{x′}⟩, ⟨S_{y′}⟩ and the vectors (a′, b′) (in the same manner as explained above for spin-1) for angles ϕ from 0^{0} to 720^{0} in steps of 90^{0}. This (especially the situation at 360^{0}) probably makes somewhat less immediate sense than the spin-1 case (but that is OK– remember this is Quantum Mechanics!) b) Optional: Contrast all this with the case of an electron originally at rest with m_{x} = \frac{1}{2} in magnetic field (0, 0, B), which is described by the time-dependent Schroedinger equation with Hamiltonian H = −M⃗ · B⃗, where M⃗ = −g(eℏ/2mc)σ⃗/2, resulting in

|ψ(t)⟩ = \frac{e^{−iωt/2}}{\sqrt{2}} (1, e^{−iωt})

Note that you are (or rather, the magnetic field is) doing an ’active rotation’ (rotating the object). Note also that a concept of ’torque’ would appear in the classical limit, much as the concept of ’force’ emerged in a classical limit of an electron deflected by the electric field.

c) A Spin \frac{1}{2} particle is prepared with spin-up in a given direction, and you measure the spin projection along a direction making an angle θ with respect to the original direction. What is the probability that you find spin-up? (We will need this when discussing the EPRB puzzle).

The atoms

7.1 Introduction

7.2 Hydrogen(like) Atoms

7.2.1 Setting up the H Equation

The Schrodinger equation of a (negative) electron of mass m_{1} “orbiting” a positive nucleus of mass m_{2} is

(−\frac{ℏ^{2}}{2m}∇^{2} − \frac{e^{2}}{4πϵ_{0}r})ψ(r⃗) = Eψ(r⃗)

where m = \frac{m_{1}m_{2}}{m_{1}+m_{2}} is the “reduced mass” of the electron, r⃗ = r⃗_{1} − r⃗_{2} is the relative distance of electron (the motion of the center of mass r⃗_{CM} = \frac{m_{1}r⃗_{1}+m_{2}r⃗_{2}}{m_{1}+m_{2}} is that of a free particle of mass M = m_{1} + m_{2}).

Unfortunately, the spherical symmetry of the potential demands that we use the spherical coordinates. Fortunately, comparison of the expressions for the Laplacian with that for L^{2} reveals that we can write the equation (ignoring various constants such as mass, ℏ, e etc)^{1} as

HΨ(r, θ, ϕ) = [−\frac{1}{r} \frac{∂^{2}}{∂r^{2}} + \frac{L^{2}}{r^{2}} − \frac{1}{r}]Ψ(r, θ, ϕ) = EΨ(r, θ, ϕ)

Since the operator L^{2} only acts on variables θ and ϕ, all three operators H, L^{2} and L_{z} commute with each other, and we can look for their common eigenfunctions

HΨ(r, θ, ϕ) = EΨ(r, θ, ϕ)
L^{2}Ψ(r, θ, ϕ) = l(l + 1)Ψ(r, θ, ϕ)
L_{z}Ψ(r, θ, ϕ) = mΨ(r, θ, ϕ)

in the form

Ψ(r, θ, ϕ) = R_{nl}(r)Y_{l}^{m}(θ, ϕ)

1the quantity m below means not the mass but the eigenvalue of L_{z}

Additional separation of variables of the ”spherical harmonics” is possible:

Y_{l}^{m}(θ, ϕ) = Θ_{l}^{m}(θ)Φ_{m}(ϕ)

7.2.2 Solving the Equation

The complete solution consists of three stages:

a) The equation for Φ_{m}(ϕ) and its solution is easy:

\frac{1}{Φ_{m}} \frac{d^{2}Φ_{m}}{dϕ^{2}} = −m^{2} Φ_{m} = e^{imϕ}

b) the equation for Θ_{l}^{m}(θ) is messy:

sin θ\frac{d}{dθ}(sin θ\frac{dΘ}{dθ}) + [l(l + 1) sin^{2} θ − m^{2}]Θ = 0

After much tedious labor it follows that

Θ_{l}^{m}(θ) = P_{l}^{m}(cos θ)

where the “associated Legendre function” P_{l}^{m}(cos θ) is defined as

P_{l}^{m}(cos θ) = (1 − x^{2})^{|m|/2}(\frac{d}{dx})^{|m|}P_{l}(x)

where the “Legendre polynomial” P_{l}(x) is defined as

P_{l}(x) = \frac{1}{2^{l}l!}(\frac{d}{dx})^{l}(x^{2} − 1)^{l}

Unfortunately, you will have to remember all of this.

Fortunately, I am just kidding. You only have to be aware of the general idea: if you really wanted to, you could obtain all these results starting from the Hilbert space properties of eigenfunctions of L^{2} and L_{z}. (as usual, it is the assigned homework that will tell you what exactly will be required ...).

c) The “radial equation” for R_{nl}(r):

\frac{d}{dr}(r^{2} \frac{dR}{dr}) − r^{2}(V (r) − E)R = l(l + 1)R

is difficult, but there is so much physics there that we cannot skip the rather involved method of solution:

1) The radial equation is greatly simplified by looking for function u(r) = rR(r). We get the equation

−\frac{d^{2}u}{dr^{2}} + [V (r) + \frac{l(l + 1}{r^{2}}]u = Eu

This looks like a simple one-dimensional equation in 0 ≤ r ≤ ∞ for the “effective potential”

V_{eff}(r) = V (r) + \frac{l(l + 1)}{r^{2}}

but the solution is still tricky:

2) specifying V (r) = −1/r, changing the variable from r to ρ = r\sqrt{−E}, and defining ρ_{0} = 1/\sqrt{−E} we get

\frac{d^{2}u(ρ)}{dρ^{2}} = [1 − ρ_{o}/ρ + l(l + 1)/ρ^{2})u(ρ)

3) We now establish the asymptotic behavior of u(ρ):

3a) for ρ → ∞ we get d^{2}u/dρ^{2} = u(ρ) so that u ∼ e^{−ρ}

3a) for ρ → 0 we get d^{2}u/dρ^{2} = \frac{l(l+1)}{ρ^{2}}u so that u ∼ ρ^{l+1}

4) We now change the definition of the function we are looking for, again, by automatically satisfying the asymptotic behavior 3a) and 3b)

u(ρ) = ρ^{l+1}e^{−ρ}v(ρ)

hoping that the remaining function v(ρ) will be simple enough to be usefully expressed as a polynomial

v(ρ) = ∑_{j=0}^{∞} c_{j}ρ^{j}

5) When we plug this into 2) we get a recursive relation for the coefficients c_{j}

c_{j+1} = \frac{2(j + l + 1) − ρ_{0}}{(j + 1)(j + 2l + 2)}c_{j}

6) and now comes the clincher: we notice that as j → ∞, u(ρ) → ∞ which is unphysical. The only way how this can be prevented is if for some j = j_{max} the series terminates: c_{j_{max}+1} = 0. From the recursive formula we see that this is equivalent to 2(j_{max} + l + 1) − ρ_{0} = 0. Finally, defining n = j_{max} + l + 1 and recalling that ρ_{0} ∼ \sqrt{1E} we get the final result:

E_{n} = E_{1}/n^{2}

and putting back all the constants (ℏ etc.) we get the energy of the ground state of the hydrogen atom

E_{1} = −\frac{mass}{2ℏ^{2}}(\frac{e^{2}}{4πϵ_{0}} )^{2} = −13.6eV

7.2.3 The energy spectrum

The spectrum of energy eigenstates and the corresponding eigenfuctions is intricate:

for n=1 we must have l=0 and m=0 so that the expansion 5) has just the first term (which is just a constant) and the only state is

(n, l, m) = (1, 0, 0)

This is the ground state with energy E_{1}.

for n=2 the spectrum is degenerate: the expansion now has the first two terms, and we get four different wavefunctions with

(n, l, m) = (2, 0, 0), (2, 1, 1), (2, 1, 0), (2, 1, −1)

all corresponding to the same energy E_{2}

for n=3 we get three terms in the polynomial, and 9 different wavefunctions with energy E_{3}

(n, l, m) = (3, 0, 0), (3, 1, 1), (3, 1, 0), (3, 1, −1), (3, 2, 2), (3, 2, 1), (3, 2, 0), (3, 2, −1), (3, 2, −2)

and so on.

7.3 The periodic Table of elements

7.3.1 Pauli

The Pauli exclusion principle allows at most one electron to be in a given state. This means that as you go from hydrogen to the heavier elements (by adding one proton and one electron at a time, plus a few neutrons as needed for the stability of the nucleus) one fills the consecutive ”shells” (note that for any given (n,l,m) there can be two differentstates corresponding to two different spin projections of the electron). Unfortunately, hydrogen is the only case we can solve analytically; for all other elements we have to use various approximation methods. A nice graphical illustration of the physics of the periodic table is on Figure.

7.4 Quantum jumps

Transitions between atomic levels n_{i} and n_{f} are accompanied by an emission of a photon of energy |E_{n_{i}} − E_{n_{f}} |. The spectrum derived above then results in the famous “Rydberg formula” found empirically in the 19th century, well before the development of Quantum Mechanics:

\frac{1}{λ} = R(\frac{1}{n_{f}^{2}} − \frac{1}{n_{i}^{2}} )

In the Grand Finale lecture, we will contemplate a fantastic quantum jump discovered by our own Physics professor Hans Dehmelt (Nobel prize 1989).

7.4.1 Atomic physics HW

Problem A: A hydrogen atom is in the state

|ψ >∼ 1.7|1, 0, 0 > −3.1i|2, 1, −1 > −1.7|3, 2, 1 > +(2.3 + 0.9i)|3, 2, −2 >

Determine the expectation values < E >, < L^{2} > and < L_{z} > Problem B: A particular transition between two shells of hydrogen atom was measured to have frequency f_{n_{i},n_{f}} = \frac{5E_{1}}{36h}. What were the initial and final n_{i} and n_{f} for this quantum jump? Problem C: Determine the nature (boson or fermion) of a) the nucleus b) the whole atom (with notation _{A}^{Z}Name) for

1) _{1}^{1}H 2) _{2}^{1}H 3) _{12}^{6}C 4) _{11}^{4}Be Problem D: Determine the wavelength of the radiation due to the jump from hydrogenic level n=4 to the level n=2. Is this transition in the visible region?

7.4.2 A glimpse at Atomic Physics

The math of the Hydrogen atom is not trivial, but the physics is very simple - the negative electron is attracted by the positive nucleus (proton, in this case), and quantum mechanical quantization keeps the system from collapsing. For multi-electron atoms, there is one substantial addition: the symmetry of identical particles. The physics is still relatively simple, but the math gets extremely difficult, and the result is astonishing: everything from stones through flowers to (possibly) human brain.

7.4.3 The spin-statistics theorem

It is a deep physics/math theorem that the wave function, upon the exchange of two identical particles, must be

symmetric if the particles have integer spin. Such particles are called “boson” and obey the “Bose-Einstein statistics”. Examples: photons, deuterons, pions and other mesons.

antisymmetric of the particles have half-integer spin. Such particles are called “fermions” and obey the “Fermi-Dirac” statistics. Examples: electrons, protons, neutrons.

62 CHAPTER 7. THE ATOMS

Bell Inequality and Beyond

8.1 Entanglement, EPRB and the Bell Inequality

A quantum mechanical system of two or more particles is said to be ”entangled” when it cannot be written as a product of the individual wave functions for the components. Thus, the state

|ψ⟩ = |m_{1} = \frac{1}{2} > |m_{2} = −\frac{1}{2} >

is not entangled, but the state

|ϕ⟩ = (|m_{1} = \frac{1}{2} > |m_{2} = −\frac{1}{2} > −|m_{1} = −\frac{1}{2} > |m_{2} = \frac{1}{2} >)/\sqrt{2}

is entangled. Such states have wonderful properties which are delightful to contemplate, and crucial for quantum computation.

8.1.1 The Bell Inequality

The simplest, and classical setup to explore entanglement if the Einstein-Podolsky-Rosen-Bohm (EPRB) experiment. A particle of spin 0 decays into two particles A and B of spin \frac{1}{2} (see Figure). Two independent observers are measuring the angular momentum projections m_{A}^{a} and m_{B}^{b} of particles A and B in directions a and b, resp.. The conservation of angular momentum implies that m_{A}^{c} = m_{B}^{c}, for any direction c. This is easy to understand: the state of the system is

|j = 0, m = 0 >=
= (|m_{A}^{c} = +1 > |m_{B}^{c} = −1 > −|m_{A}^{c} = −1 > |m_{B}^{c} = +1 >)\sqrt{2} =
= (|m_{A}^{d} = +1 > |m_{B}^{d} = −1 > −|m_{A}^{d} = −1 > |m_{B}^{d} = +1 >)\sqrt{2} =
= ....

(note that +1 represents +\frac{1}{2} and -1 represents −\frac{1}{2}).

A decision by either observer to measure along a given direction. and finding a given value brings the whole two-particle system into the corresponding state, and insures that the other observer finds the opposite value, if he measures along the same direction. This is a peculiar ”action at a distance”, and although it does not violate the Letter of Relativity, many people felt (and some still do) that it violates the Spirit of Relativity. The perfect anti-correlation, in each individual decay, of the spin projections in respect to some (but otherwise arbitrary) direction discourages one from thinking that the result of the measurement is somehow only determined by the interaction of the system with the measuring apparatus. One way to save both the Quantum Mechanics and the ”Common Sense” might seem to be the following: imagine that for any given decay, all the angular momentum projections, in all possible directions, really exist, and satisfy m_{A}^{a} = −m_{B}^{a} for any direction a. However, in the manner similar to the measurement of the momentum following a measurement of the position, a measurement of the spin projection along a particular direction disturbds the state, making the measurements along two different directions on the same particle mutually incompatible (recall that the spin projection operators do not commute with each other ...). At this point, the reader is invited

a) to realize how nice that would be. It would reduce the mysterious correlation to something we can ”understand”. Suppose I have a black and a white marble in my pocket. Without looking, I put one in the pockets of my two friends. When one of them looks and finds a white marble, he can be sure that the other one will find a black one. You can entertain yourself with elaborating on this example - the point is that the properties really were there all along ...

b) to invent some other ’explanation’ of the perfect anti-correlation along an arbitrary direction.

Unfortunately (or fortunately, depending on your point of view) it cannot be so. Amazingly, John Bell has found an experimental consequence which would follow from the ability to even think about the projections along different directions. Consider the observer A having a choice between directions a and a′, while observer B will decide between directions b and b′. Although one cannont measure all four quantities m_{A}^{a}, m_{A}^{a′},m_{B}^{b} and m_{B}^{b′} in one particular decay, one can, according to the assumption, think about them. One such thought is that for each decay

|m_{A}^{a}(m_{B}^{b} + m_{B}^{b′}) + m_{A}^{a′}(m_{B}^{b} − m_{B}^{b′})| = 2

Consequently, the average (expectation) values must satisfy the inequality

| < m_{A}^{a}m_{B}^{b} > + < m_{A}^{a}m_{B}^{b′} > + < m_{A}^{a′}m_{B}^{b} > − < m_{A}^{a′}m_{B}^{b′} > | ≤ 2

This is the celebrated Bell Inequality (actually derived in this particular form by Clauser, Horne and others), and all the ingredients are experimentally measurable. Moreover, the prediction of any possible ”locally realistic” theory are easy to find: take all direction to be coplanar, with a = b and other directions as whown on Figure. Define the correlation

C(θ) ≡< m_{A}^{a}m_{B}^{b} >

where θ is the angle between a and b. By symmetry, any theory must have

C(0) = −1 c(90) = 0 C(45) = −C(135)

The Bell Inequality then implies

−0.5 ≤ C(45) ≤ +0.5 − 0.5 ≤ C(135) ≤ +0.5

What about Quantum Mechanics? Recall that the probability of finding the particle with spin up in a given direction, when it was prepared with spin up in direction making an angle θ with the direction of the measurement

| < up rotated by θ|up > |^{2} = (1 + cos θ)/2

HW: fill in the missing steps. Obviously, Quantum Mechanics is in disagreement (see Figure) with any theory which would allow one to even think about several spin projections existing simultaneously! This cannot but remind you of the famous Goedel Undecidability Theorem ... And what about Nature? The experiment has been performed (see Aspect et al., PRL 49, 1804 (1982) and subsequent experiments) and result violate the Bell Inequality (and agree with Quantum mechanics).

8.2 Quantum Computing

After decades of pondering the quantum puzzles, without much progress, people decided to simply accept the quantum formalism, and exploit it technologically. Quite suddenly, a whole industry of ”Quantum Information” was born, with Quantum computing, Cryptography, “Teleportation” (of quantum state only!) and more. Some of these applications are mindboggling. As an example, we can now scan a picture without shining any light (or anything else) on it! For an illustration of the basic quantum computing idea see Fig

8.3 The Grand Finale

Lesson from Bell Inequality, as expressed by David Mermin:

“The point is no longer that quantum mechanics is an extraordinarily (and, for Einstein, unacceptably) peculiar theory, but that the world is an extraordinarily peculiar place.”

The lesson, as expressed by Vladi in the Introduction to his “Psychology of Quantum Mechanics” (work in progress):

This is a tentative collection of notes on events, some well known, some less so; some old, others recent; some serious, some funny (and some both) - all illustrating that Quantum mechanics is, after all, product of human effort. Some tell us about the strange ways progress is often made (often in the manner of sleepwalkers as Arthur Koestler and then John Bell so beautifully described), some show how far are people willing to go to defend their favored theory, and some showing the limitations of human language when it comes to describing the quantum phenomena - the whole range of human greatness and folly.”

And you now know something about Hilbert space, angular momentum, hydrogen atom, entanglement, superposition of states and Quantum Physics in general!

66 CHAPTER 8. BELL INEQUALITY AND BEYOND

Chapter 9 Figures

Figure 9.1: Caption for singleslit 68 CHAPTER 9. FIGURES Figure 9.2: Commercially available blackbody. With a heater (or cooler) and temperature lock.

Figure 9.3: Horizontal is the wavelength in mm. The UV catastrophe is obvious. To appreciate the low frequency behavior see the LogLog plot next.

Figure 9.4: LogLog of previous plot. I am mesmerized by this graph :-)

70 CHAPTER 9. FIGURES Figure 9.5: An elementary integral/sum lesson.

Figure 9.6: The error bars are smaller than the thickness of the line. Magnified at the bottom; the curves are no longer relevant.

72 CHAPTER 9. FIGURES Figure 9.7: The Nobel for COBE for this

73

Figure 9.8: Chronology of the Big Bang Figure 9.9: Planck spectra for three temperatures. Horizontal scale is CMBR wavelength in mm; note the absolute vertical scale! The legends give Temp of CMBR, time since BB and energy of the CMBR

75

Figure 9.10: Cartoon of the surface of last scattering 76 CHAPTER 9. FIGURES Figure 9.11: The three spacecraft experiments. COBE resolution is barely sufficient to see the first peak (see Figure 9.14); the Planck resolution cannot be improved (see Figure 9.13)

77

Figure 9.12: Many interesting facts about the first million years.

78 CHAPTER 9. FIGURES Figure 9.13: The definitive acoustic spectrum from Planck. The error bars of the residuals are multiplied by 10 on the right of the dashed like, Figure 9.14: The preWMAP (i.e. also prePlanck) spectrum from 1999. The solid line is the prediction they call ”the standard model”. Compare with the previous plot. Those theorists in 1999 were clairvoyant(s)!!!(see text on the bottom of p.6 )

80 CHAPTER 9. FIGURES Figure 9.15: Caption for FeynmanCentralMystery

Figure 9.16: Caption for beamsplitter

Figure 9.17: The double slit setup with the lower slit blocked. You hear clicks as electron arrive at the detector.

82 CHAPTER 9. FIGURES Figure 9.18: You OPEN previously closed slit, and the electrons STOP arriving Figure 9.19: Now you close the top slit, and electrons are arriving again - click, click clikclik ...

84 CHAPTER 9. FIGURES Figure 9.20: Caption for 2MS Figure 9.21: Caption for spectrometer

Figure 9.22: Caption for phases 86 CHAPTER 9. FIGURES

Figure 9.23: Caption for 2MSphases

87

Figure 9.24: Caption for MZfringes 88 CHAPTER 9. FIGURES

Figure 9.25: Caption for qpix Page 1

89

Figure 9.26: Caption for qpix Page 2 90 CHAPTER 9. FIGURES

Figure 9.27: Caption for qpix Page 3

91

Figure 9.28: Caption for qpix Page 4 92 CHAPTER 9. FIGURES

Figure 9.29: Caption for Afshar

93

Figure 9.30: and the stop again when they have TWO paths open instead of just one ...

94 CHAPTER 9. FIGURES Figure 9.31: Caption for fringes

95

Figure 9.32: Caption for PhotoeffectCombo 96 CHAPTER 9. FIGURES

Figure 9.33: Caption for SternGerlach Figure 9.34: Caption for Heisenberg1

97

Figure 9.35: Caption for tunneling 98 CHAPTER 9. FIGURES

Figure 9.36: Caption for WavePacketComplexScrew

99

Figure 9.37: Caption for AtomicGraph2 100 CHAPTER 9. FIGURES

Figure 9.38: Caption for barr Figure 9.39: Caption for Bell1 102 CHAPTER 9. FIGURES

Figure 9.41: Caption for Bell3

103

Figure 9.42: Caption for bellfigs Page 1 104 CHAPTER 9. FIGURES

Figure 9.43: Caption for box

105

Figure 9.44: Caption for ComplexDefinitions 106 CHAPTER 9. FIGURES

Figure 9.45: Caption for PhotoeffectCombo

107

Figure 9.46: Caption for PhotoeffectCombo 108 CHAPTER 9. FIGURES

Figure 9.47: Caption for CoalVsNuclear

109

Figure 9.48: Caption for bellfigs Page 1 110 CHAPTER 9. FIGURES

Figure 9.49: Caption for SymmetryForces Figure 9.50: Caption for Bell1 112 CHAPTER 9. FIGURES

Figure 9.52: Caption for Bell3

Physics and Math Appendix

10.1 Math and Physics Survey and Warm-up

This chapter covers some basic knowledge useful for a full understanding of the book. The subsequent chapters contain considerations of similar, but not greater, difficulty, so the “warm-up” serves also as an assurance that the reader will be able to understand the book. As promised, they are no real pre-requisites that would not be covered by a good high school. Therefore, everything more advanced will be presented “from scratch”. Even so, it will be no easy reading for some readers, so patience and perseverance will be required - and, I promise, richly rewarded. Already the very next section will demonstrate how the simplest possible math consideration may quickly lead to fascinating applications.

10.1.1 Scientific Notation and a Glimpse at Infinity

In any discussion of Nature, we need to be able to deal with an extremely large range of physical quantities. For some strange reason, most authors of popular books seem convinced that the general public is not capable of comprehending a simple system of powers of 10 known as the “scientific notation”. Even more bizarre is the apparent belief that expressing large numbers in the “ordinary way” such as “1,000,000,000,000,000,000” or “one hundred billion billion billion” provides any useful information at all. In this book, we use the scientific notation frequently and consistently. Readers who never encountered this reasoning should be prepared to spend some time digesting and mastering this topic: please be assured that it is most definitely worth your while! And even physics majors - who may be bored by the apparent trivialities that begin this section - usually find the end of the section highly non-trivial and interesting.

The whole affair starts very simply indeed:

10^{2} = 10.10 10^{3} = 10.10.10 10^{5} = 10.10.10.10.10

and so on. Inversely:

10^{−2} = \frac{1}{10.10}  10^{−3} = \frac{1}{10.10.10}  10^{−5} = \frac{1}{10.10.10.10.10}

It is easy to see that each additional power of 10 means adding a zero, so that the large number 1,000,000,000,000,000,000 in the first paragraph is nothing but simply 10^{18} (just count the zeros). It is also easy to see that multiplying two powers of ten corresponds to simple addition of the exponents: for example

10^{2}.10^{3} = (10.10).(10.10.10) = 10^{2+3} = 10^{5}

Therefore, the second large number in the first paragraph above (one hundred billion billion billion) is equal to 10^{2}.10^{9}.10^{9}.10^{9} = 10^{2+9+9+9} = 10^{29}. For numbers smaller than 1.0, it is equally easy to see that it is the position of the first non-zero digit after the decimal point which gives the correct (negative) power of 10, so that for example: 10^{−3} = 0.001 = one thousandth, 10^{−6} = 0.000, 001 = one millionth and so on. And finally, any division by a power of 10 corresponds to subtraction of the powers, as in

10^{6}/10^{2} = \frac{10.10.10.10.10.10}{10.10} = 10.10.10.10 = 10^{4} = 10^{6−2}

We can summarize our results so far

10^{2}10^{3} = 10^{2+3} = 10^{5} \frac{10^{6}}{10^{2}} = 10^{6−2} = 10^{4}

by two general equations: for an arbitrary choice of the values of m and n we have:

10^{n}10^{m} = 10^{n+m} \frac{10^{n}}{10^{m}} = 10^{n−m}

Note also that just a moment of reflection will convince you that

(10^{n})^{m} = 10^{n.m} e.g. (10^{2})^{3} = 10^{2}.10^{2}.10^{2} = 10^{2+2+2=10^{2×3}} = 10^{6}

As an exercise, you should be able to evaluate

\frac{2 × 10^{12} × 6 × 10^{−9}}{3 × 10^{−7}} =

and express in plain English the numbers:

\frac{10^{4}}{2} =  and  \frac{10^{−2}}{2} =

Also: most students know that 10^{0} = 1 but few can explain why is it so. Yet, you are now equipped to show this from the simple relations we just introduced.

The solutions of this (and other) “homework” exercises are in Appendix X at the end of the book. However, I advise all readers to spend at least a few minutes on each problem trying to solve it, before turning to the solutions.

10.1.2 Nicknames for some standard exponents

Some powers of 10 have nicknames (see below) but for really large or really small numbers there are no names - just the scientific notation.

Standard names and prefixes:

factor  name  prefix  example
10^{12} trillion  Tera  TeV (tera-electronVolt)
10^{9}  billion  Giga  GW (gigawatt)
10^{6}  million  Mega  MΩ (megaOhm)
10^{3}  thousand kilo  kg (kilogram)
10^{−1} tenth  deci  dl (deci-liter)
10^{−2} hundredth centi  cm (centimeter)
10^{−3} thousandth milli  ms (millisecond)
10^{−6} millionth  micro (μ ) μ g (microgram)
10^{−9} billionth nano nm (nanometer)
10^{−10}  Angstrom (A)
10^{−12} trillionth  pico  ps (picosecond)
10^{−15}  femto  fm (1 Fermi = femtometer)

10.1.3 Note on units

We will consistently use the metric system, with MKSA units of length (m), mass (kg)[sic], time(s) and current (A). Some derived metric units are: work/energy: Joule (J), power: Joule/s = Watt, force: kg m/s^{2} = Newton (N), pressure: N/m^{2} = P ascal(P a)

Note that some dimensionless quantities have been given names, too (dB, cents, Radians, ...)

10.1.4 A glimpse at Infinity

The usefulness of the scientific notation follows from the ease with which we can represent the huge ranges of various quantities mentioned above. So for example, we will soon discuss the range of distances from 10^{−34} meters (the “Planck scale”) all the way to 10^{26} meters (the distance to the edge of the observable Universe). So if you start from the Planck scale and increase it million times, your get 10^{−34}10^{6} = 10^{−28}. If you take this distance and increase it billion times, you get 10^{−28}10^{9} = 10^{−19} - still a ways from 10^{+26}! There is an anecdote that nicely illustrates the enormous ranges we deal with. An article on cosmology in a scientific journal was followed by an Errata in a subsequent issue, apologizing for an inadvertent error: the main result was off by a factor of million[sic]. However, the Errata says, this does not change any conclusions of our paper. And now, to introduce a really large number, I should mention that a well known software company is named after a misspelled nickname for 10^{100} which is called a googol: it is a 1 followed by one hundred zeros. This is a number much, much larger than the number of seconds elapsed since the Big bang (which is only about 10^{17}). You see that a number billion times smaller or billion times larger than a googol is 10^{91} or 10^{109} – who cares that it is “not exactly 10^{100}”.

Now, if you have a tendency for mischief, you might think: what about 10^{googol} ? That would be 10^{10^{100}}: a 1 followed by a googol of zeros^{1}. As you can read in Wikipedia, this number could not be printed even if all matter in the visible Universe was converted into paper and ink. Curiously, such a monster is important enough to have its own name: it is called a “googolplex”, and even more curiously, it is (possibly) useful to think about numbers as large as this, as we will now see.

The part of Universe visible to us is called a “Hubble volume”, and it is some 10^{27} meters across. In a serious (it is claimed) scientific paper, a calculation is presented showing that if the Universe is infinite or just very, very large, then we can expect, at a distance of some 10^{10^{115}} meters from us, A Hubble volume identical in all details to our Hubble volume. This means that in “that Universe” there is a copy of you reading this book and thinking what you are thinking^{2}. Now, to get a glimpse at infinity, express that distance in terms of the size of ”our Universe”, i.e. how many of our Hubble volumes we would have to put next to each other to get to that place. It is a straightforward consequence of our elementary introduction to the scientific notation that instead of

10^{(10^{115})}

the answer is

10^{(10^{115}−27)}

Now try to visualize the difference between 10^{115} and 10^{115}−27! For “all practical purposes”, they are the same. This means that the distance to that ”mirror” Universe is, for all practical purposes, 10^{10^{115}}, independent of whether you measure it in inches, meters, light years or in Hubble volumes! Billion billion billion times smaller or billion billion billion times larger - does not make any difference.

You may have to spend a few minutes to fully comprehend how incomprehensible this result is. And if you do, you will get a glimpse at the nature of infinity. Galileo, who was not exactly the most modest of scientists, was asked once if he thought the Universe was finite or infinite. He replied, with uncharacteristic modesty, that either possibility was beyond his imagination.

1Please note that 10^{10^{100}} is defined as 10^{(10^{100})} which is different from (much larger than) (10^{10})^{100} = 10^{10.100} = 10^{1000} - the first number has a googol of zeros; the second has “just” 1000 zeros (it is still an obscenely large number)

2The article begins like this: “Is there another copy of you reading this article, deciding to put it aside without finishing this sentence while you are reading on? A person living on a planet called Earth, with misty mountains, fertile fields and sprawling cities, in a solar system with eight other planets. The life of this person has been identical to yours in every respect ...

You probably find this idea strange and implausible, and I must confess that this is my gut reaction too.

Yet it looks like we will just have to live with it, since the simplest and most popular cosmological model today predicts that this person actually exists ...”

Curiously, it is quantum mechanics that allows this calculation to be made - Universe with continuous spectra of physical quantities would be much more complicated than our quantum Universe. However, in quantum physics many quantities can only occur at discrete, “quantized” values, and that makes the nuber of possible combinations finite (although still extremely large. Then an arbitrary combination that is possible, such as for example everything in our Hubble volume, must occur somewhere else.

10.1.5 A case study in modern cosmology

In 2002, the cosmologist Andrei Linde (from the Department of Physics, Stanford University) published a paper on ”Inflation, Quantum Cosmology and the Anthropic Principle” ^{3} . In it, he calculates that after an inflation lasting 10^{−35} seconds, starting from the Planck size l_{p} ∼ 10^{−33} cm, the Universe grew to the size l ∼ 10^{10^{12}} cm. In 2006, Linde gave a review talk on “Inflationary Cosmology”. In it, he calculates that after an inflation lasting 10^{−30} seconds, starting from the Planck size l_{p} ∼ 10^{−33} cm, the Universe grew to the size l ∼ 10^{10^{10}} cm. After this calculation, both papers continue with an identical text: “This number is model-dependent, but in all realistic models the size of the Universe after inflation appears to be many orders of magnitude greater that the size of the part of the universe which we can see now, l ∼ 10^{28} cm. This immediately solves most of the problems of the old cosmological theory.”

HW: Contemplate these numbers, appreciating the duration of the inflation and the initial and final sizes. Then determine

a) what is the difference (rather: what is the ratio) between the two sizes of inflated Universe above (i.e. what is the error in one of the papers, if the other one is in fact correct).

b) what is the numerical result for the ”many orders of magnitude” in the text above.

10.2 Exponential growth and Exponential decay

10.2.1 Exponential growth

It is easy to see that if after some “doubling time” T the population of a given sample doubles, then after two doublings the population quadruples, after 3 douling times in increases by 2.2.2 = 8, so in general after n doublings we end up with 2^{n} times increase of the original population.

3Inflation is a process similar to inflating a baloon, postulated to occur at the very beginning of the Big Bang. It ’explains’ why today the apparently causally disconnected regions of the Universe seem homogeneous. Dr. Linde is the author of the “eternal chaotic inflation” scenario, in which the birth of new Universes and their subsequent inflation happens frequently, randomly, like bubbles in boiling water.

Quantum cosmology applies Quantum Mechanics to the Universe as a whole, bravely ignoring the role of an observer in quantum mechanics. The Many Worlds interpretation of the theory postulates that whenever there is a quantum choice, all possible choices actually happen: each time the Universe splits into “parallel Universes”.

The Anthropic Principle ’explains’ the apparent ’fine tuning’ of constants and laws of Nature by pointing out that we - necessarily - live in a Universe which has the constants making intelligent life possible.

10.2.2 Exponential decay

Similarly: if after some “halving-time” T the population of a given sample decays (reduces) to one half, then after n halving times we and up with 1/2^{n} = 2^{−n} decrease. All this is sufficiently simple to just verify your understanding by a few exercises:

1) An E.coli bacterium will reproduce (split/double) every 20 minutes (given adequate food ...). Determine the bacterial mass after 2 days of this, starting from a single bacterium (one E.Coli bacterium has mass of 10^{−15} kg).

2) For fun, do the Chinese chess board/rice problem: the Emperor promised his court scientist to put 1 grain of rice on the first field, 2 grains on the next, then 4 grains, 8 grains etc. - how many tons of rice did the Emperor have to give to the scientist? (and please explain how you estimated the mass of one grain of rice).

And to consider an application of great importance, consider this:

3) Isotopic analysis of ancient bones yielded a ratio of Carbon-14/Carbon-12= 2.5 × 10^{−13}. How old are the bones? (input: the ratio in living tissues is 10^{−12}, the intake of Carbon-14 stops at death, and the half-life of Carbon-14 is 5,730 years.)

Note: in the same manner, many other pairs of isotopes are used for radiometric dating of samples.

10.3 Ranges of selected physical quantities:

10.3.1 Mass (in kilograms - kg)

10^{−30} electron
2 × 10^{−27} proton
10^{−15} virus
10^{−10} speck of dust
6 × 10^{24} Earth
2 × 10^{30} Sun
10^{43} Galaxy
10^{53} observable Universe

10.3.2 Energy (in Joules - J)

10^{3} lift 100 kg by 1 m
3.6 × 10^{5} 100W bulb left on for 1 hour

10^{7} chemical oxidation (1kg of (fuel+oxygen)). Fuel may be Carbon/Hydrogen/gasoline/food(fat)/... Note: 1 kg TNT = 4.3 × 10^{6}J

7 × 10^{13} complete fission of 1 kg of pure ^{235}U (average over modes)
3 × 10^{14} complete fusion of 1 kg of pure _{1}^{2}H (deuterium) (average over modes)
4.3 × 10^{13} Hiroshima bomb (10 kiloton TNT)
2.6 × 10^{17} most powerful weapon ever tested (60 Megaton TNT equivalent)
9 × 10^{16} complete annihilation of 1 kg (matter+antimatter)
4 × 10^{26} energy emitted by the Sun every second

10^{43} energy emitted in a Gamma Ray Burst (lasting for a second or so ... - for its duration it outshines the rest of the Universe!)

10.3.3 Time (in seconds - s)

10^{−43} Planck time - we have no idea what a shorter time might mean
10^{−23} lifetime of least-stable particles
10^{−15} shortest light pulse produced
10^{−12} typical molecular rotation period
3 × 10^{7} one year
10^{17} age of Universe
3 × 10^{27} lifetime of ^{82}Se
10^{40} lower limit on proton lifetime

10.3.4 Distance (in meters - m)

10^{−34} Planck length; we have no idea what a shorter distance might mean
10^{−18} we know that electron is not larger than this ...
10^{−15} proton radius
10^{−10} atomic radius
10^{−6} size of a typical virus
10^{−4} thickness of this page
10^{4} Mt. Everest
6 × 10^{6} radius of Earth
4 × 10^{8} Earth to Moon distance
7 × 10^{8} radius of Sun
1.5 × 10^{11} Earth to Sun distance
10^{16} a light year: the distance traveled by light in one year
4 × 10^{16} nearest star (Proxima Centauri)
2 × 10^{20} distance to Galactic center
10^{26} distance to the edge of observable Universe

10.3.5 Scales of the Universe

In everyday life, our senses give us the impression that there is a lot of “stuff” - hard stuff you can sit on, liquids you can drink or swim in, air you can breathe, stars you see. But the Universe, when seen both at the atomic as well as at the cosmic scales, is almost completely empty:

10.3.6 Atomic structure of matter

As we know, all matter is composed of atoms: nuclei surrounded by orbiting electrons. Typical nucleus is about 10^{−15} m, in diameter, the resulting electron cloud Suppose the nucleus is a grain of sand 0.1 mm in radius. Then the electron(s) - (nearly) massless and (probably) pointlike - orbit about 10 m away; the next nucleus is about 20 m away. This is the structure of the chair you are sitting on, of an iron brick or of a gold ring.

10.3.7 solar system scale:

Imagine now that the Earth is as a speck of a small grain of sand. Then the Sun is a little ball of about an inch in diameter, and Earth orbits at the distance of 2 meters, Jupiter is a large grain of sand orbiting at 11 meters, and the Solar system ends at 100 meters.

10.3.8 Galactic scale

Using the same scale as for the solar system (above), the nearest star would be at 1,000 km.

10.3.9 Intergalactic/cosmological scale

The ultimate dilution of matter is provided by the distances of millions of light-years between galaxies. In addition, recent observations show huge voids - empty “bubbles”, with no galaxies within the diameters up 100 million light years.

10.3.10 Scale of densities of matter:

We express densities of powers of ten (ρ = 10^{N}kg/m^{3}) and we tabulate the values of the exponent N for different media:

  N    medium
------------------------------------------------------------------------
 17  a nucleus, or a neutron star
 13  Universe at end of the ``electroweak epoch"
  4  Pb (Lead)
  3  water
  0  Earth atmosphere
-18  best laboratory vacuum
-27  the average Universe as a whole (including all stars, planets etc.)
-------------------------------------------------------------------------

Just let these number sink in, and observe:

A spoonful (about 10^{−6} m^{3} with neutron star density has mass of about 10^{−6}.10^{17} = 10^{11} kg, i.e. 100 million metric tons!

At a human scale: a spoonful of water has the mass of 10^{−6}.10^{3} = 10^{−3} kg, i.e. one gram

And at the other extreme, the best vacuum we can produce in the laboratory contains some 10^{−18}.10^{−27} = 10^{9} atoms per cubic meter (recall that a Hydrogen atom has a mass of about 10^{−27} kg). However, the Universe as a whole is much emptyer that that. If we could grind up all its constituents (stars/planets etc) and distribute the resulting atoms evenly through the Universe, we would have just 10^{−27}.10^{27} which is just one atom in each cubic meter, compared to a billion atoms per cubic meter in our best vacuum!

These considerations should not be dismissed as pedantic, boring numerology. The Universe really is almost empty, and we really are an infinitesimal part of a infinitesimal part of the Universe.

And yet ....

10.4 Math Overview

10.4.1 Calculus

Familiarity with calculus is assumed. It is not a pre-requisite, since the amount of calculus needed in this course can be refreshed, or even learned from scratch, in a few days. In my experience of teaching quantum physics for many years, the math is not the problem; the real, quite difficult problem is to be able to cope with physics concept at variance with our everyday experience.

10.4.2 Differentiation

Start with a full realization that the derivative is a slope. Most rules follow from this almost immediately: derivatives of a product/ratio (HW: prove the rules from first principles); exponential and powers; function of a function; etc. You need to be able to use the l’Hospital rule (dating from 1696!) to deal with 0/0 and ∞/∞. And we will need the Taylor expansion of function f(x) around the point a:

f(x) = ∑_{n=0}^{∞} \frac{f^{(}n)(a)}{n!}(x − a)^{n}

Keeping only the first m + 1 terms is called ”working in the m-th approximation”. HW: To get a feeling for this important concept, show that in the first approximation

\sqrt{1 + x} = 1 + x/2

Do this with and without the use of the Taylor expansion.

10.4.3 Integration

Integration tends to be considerably more difficult that differentiation, since we cannot apply “mechanical” rules here, and a lot of experience is needed. This course is not meant to teach you that, so for anything more complicated thn simple variable substitution, or integration by parts, we will use tables of integrals. However, it will be very important for us to fully understand the connection between differentiation and integration, i.e. the ”fundamental theorem of calculus” (HW: ”prove” it from first principles”). The HW problem on the Planck discovery of the quantum is a beautiful example of this.

10.5 Complex Numbers:

The theory of the functions of complex variables is not simple, but it also is very powerful and very beautiful. (Un)fortunately, we will only need the simplest complex number concepts:

We define a complex number z

z = x + iy = Re(z) + i.Im(z) = a(cosϕ + i.sinϕ))

where i^{2} = −1 - see Figure 10.1

10.5.1 Feynman’s “Amazing Jewel”

Chapter 22 of Volume I of the celebrated Feynman Lectures begins by stating:

“In our study of oscillating systems we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics ... an amazing jewel ...”

I cannot resist from sharing some of my admiration for Feynman genius and wisdom, as well as his remarkable skills in manipulating numbers. So here is a simplified version of his brilliant celebration of the power of human reasoning – normally one does this proof using calculus. Feynman does it “from first principles”:

Consider the function f(x) = e^{x} where “e” is the base of the natural logarithms e=2.71828... . Using your calculator you can easily confirm, to arbitrary precision, that for very small values^{4}

f(d) = e^{d} = 1 + d

Since any number D that is not small can be expressed as sum of many small numbers: D = d+d+d+d+....d, and since e^{a+b} = e^{a}e^{b}, we get

e^{D} = e^{d}e^{d}e^{d}...e^{d} = (1 + d)(1 + d)1 + d)...(1 + d)

The first three columns on Figure 10.2 show a simple EXCEL calculation demonstrating this.

Complex numbers obey all the usual rules (as long as you always substitute i^{2} = −1).So assume that the above equation is also true for d = iϕ where ϕ is very small. Then for D = iΦ when Φ is not small, we can follow the above logic, split Φ into N pieces all equal to ϕ = Φ/N and we get

e^{iΦ} = e^{iϕ}e^{iϕ}e^{iϕ}...e^{iϕ} = (1 + iϕ)(1 + iϕ)(1 + iϕ)...(1 + iϕ)

and we can easily evaluate this for any value of the argument Φ (again, always remembering that i^{2} = −1). The results of the corresponding EXCEL calculation is on Figure 10.2, and you can check, to arbitrary precision, that the real part is nothing but cos Φ and imaginary part is sin Φ, so that

e^{iΦ} = cos Φ + i sin Φ

for any value of Φ (see Figure 10.3)^{5}.

This is the promised equation (usually called the Euler formula): spectacular from many points of view, and ubiquitous in mathematics, physics, signal processing and just about any branch of the “hard” sciences.

10.5.2 Some applications of the Euler formula

Previously, we used the ”Cartesian” representation

z = x + iy

\frac{df_{1}(x)}{dx} = \frac{e^{x^{1/2}}}{2\sqrt{x}}

4Hint: start by using d = 0.1 and you will get e^{d} = 1.105.... This differs from 1 + d just by 0.5% - good but not good enough for our use. So try d=0.01, 0.001, 0.0001 and see what happens).

5For the bottom part of the Figure 10.3 the simplest choice f_{1}(x) = e^{x^{1/2}} does not work because the derivative at x=0 is singular:

The Euler formula provides the “polar” representation

z = aeiϕ

The Cartesian form is useful for linear operations (e.g. z_{1} + z_{2} = (x_{1} + x_{2}) + i(y_{1} + y_{2}) etc.), while the polar form is useful for multiplication, division and exponentiation (e.g. z_{1}/z_{2} = (a_{1}/a_{2})e^{i(ϕ_{1}−ϕ_{2})}, \sqrt{z} = \sqrt{a}e^{iϕ/2} etc.). Notes:

1. A very useful expression that immediately follows is i = e^{iπ/2}. That can be used to determine, for example

\sqrt{i} = \sqrt{e^{iπ/2}} = (e^{iπ/2})^{1/2} = e^{iπ/4} = (1 + i)/\sqrt{2}

2. Even more fun is

i^{i} = (e^{iπ/2})^{i} = e^{−π/2}

which means that

i^{i}e^{π/2} − 1 = 0

Just about all mathematical symbols, some seemingly not related, are connected in this single expression^{6}!

3. A less fanciful but much more useful example is a trigonometric application:

e^{i(α±β)} = cos(α ± β) + i sin(α ± β) = e^{iα}e^{iβ} = (cos α ± i sin α)(cos β ± i sin β)

Equating the both sides of the real and imaginary parts of the second expression with those of the 4th expression, you get some very useful trigonometric formulas. Just try to prove them without complex numbers (the way the ancients did it :-)

4. complex conjugation:

z^{∗} = x − iy = ae^{−iϕ} |z| = \sqrt{z.z^{∗}} = a = \sqrt{x^{2} + y^{2}}

5. Note the origin of the Euler formula in the expansion

e^{b} = 1 + b + b^{2}/2! + b^{3}/3! + b^{4}/4! + b^{5}/5! + ...

for b = iϕ.

HW: Determine real and imaginary part of \sqrt{i}, i^{i} (proving the remarkable equation i^{i}e^{π/2} − 1 = 0), and derive the expressions for sin(α + β) and cos(α + β)

6We should note that our expression is just one of infinitely many that follow from the fact that the \sqrt{−1} can also be written as i = −e^{i2π/2}, i = e^{i3π/2}, i = −e^{i4π/2} etc.

10.6 The Dirac Delta function

Consider a limiting process in which a function which peaks at a given position (say at x=a) gets narrower and narrower while keeping its integral (i.e. the area beneath the graph) equal to 1 - see Fig. 4. In the limit, this becomes a remarkable ”function” δ(x − a) which is equal zero everywhere except at x=a, and yet

∫ f(x)δ(x − a)dx = f(a)

for arbitrary function f(x) - see Fig. 5. Mathematicians call such a creature a ”distribution” or a ”functional”, but we will call it the ”Dirac delta-function” - it is ubiquitous in many Physics applications. The delta function is the limit of various peaked functions - some of the many possible ’representations’ are graphical, some are analytical. For our use, the most useful analytical form will be

δ(x) = (1/2π) ∫_{−∞}^{∞} e^{ikx}dk = 1/2π) lim_{L→∞} ∫_{−L}^{L} e^{ikx}dk ∼ lim_{L→∞} L\frac{sin Lx}{Lx}

HW: fill-in the missing steps (hint: see Figure 10.4) Notes:

the Dirac δ function is not square-integrable:

∫ δ(x)dx = 1 but ∫ δ^{2}(x)dx = δ(0) = ∞ !

The resulting normalisation problems, when not trivial, are resolved by using wave packets, or by using a box-normalisation with large but finite L.

In order to fully appreciate the defining property of the δ function:

∫ f(x)δ(x − y)dx = f(y)

it is useful to compare it with the analogous property of the discrete case:

∑_{n} f_{n}δ_{nk} = f_{k}

with the Kronecker delta defined as δ_{mn} = 0 for m ≠ n, δ_{mm} = 1.

Many other properties of the delta-function (see text A.2) are easily demonstrated using the behavior of the defining integral.

10.6.1 The Fourier Transform

Define

F (k) = (1/\sqrt{2π}) ∫ f(x)e^{−ikx}dx

Then it is easy, using our newly acquired facility with the δ-function, to see that

(1/\sqrt{2π}) ∫ F (k)e^{ikx}dk = f(x)

These two expressions form the famous Fourier transform and its inverse (you are probably more familiar with Fourier transform with variables (t, ω) instead of (x, p) - it is exactly the same thing ...).

10.6.2 The Fourier Series

When the ”spectrum” F (k) happens to be a sum of discrete, equidistant peaks:

F (k) = ∑_{−∞}^{∞} \sqrt{2π}a_{n}δ(k − nk_{1})

it is easy to see that the resulting f(x) will be periodic with period 2L = 2π/k_{1}, described by a Fourier series:

f(x) = ∑_{n=0}^{∞} A_{n}cos\frac{nπx}{L} + ∑_{n=1}^{∞} B_{n}sin\frac{nπx}{L} = ∑_{−∞}^{+∞} a_{n}e^{inπx/L}

We see that Fourier series is a special case of Fourier transform (also in the opposite direction: any periodic function can be shown to have spectrum consisting of equidistantly-spaced delta-functions ...).

The ‘base’ functions e^{inπx} satisfy an obvious (and very useful) orthonormality condition:

\frac{1}{2L} ∫_{−L}^{+L} e^{inπx/L}e^{−imπx/L} = δ_{mn}

This enables a determination of the coefficients: multiplying the expansion of f(x) by e^{−imπx/L}, and integrating over x, one gets

a_{m} = \frac{1}{2L} ∫_{−L}^{+L} f(x)e^{−imπx/L}

Remarkably, we can also show that Fourier transform is a special case of Fourier series:

Define

k = πn/L F (k) = \sqrt{\frac{2}{π}} La_{n}

and make transition L → ∞ (i.e. Δk = π/L → dk, and replace the sum by an integral):

f(x) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{+∞} F (k)e^{ikx}dk

The inverse transform follows from the expression for a_{n}:

F (k) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{+∞} f(x)e^{−ikx}dx

10.7 Mandelbrot Set

In our final warm-up we will illustrate the use of the complex numbers on the famous Mandelbrot set. Most people have seen the spectacular images, and many sleepless nights have been spent with one of the many free browsers exploring the limitless zooming into the set. But very few really understand the math, and implications of all this. As I say to students: this will not necessarily improve your grades or job prospects, but it will make you a better-educated person.

The Mandelbrot set has been called the most beautiful set in all of Mathematics. It is remarkable by the utter simplicity of its definition, compared to the infinite complexity of the resulting pattern. We will use this set as a metaphor for the concept of emergent complexity, as an illustration of the nature of the laws of physics (including the possible(?) Theory of Everything), and we will compare its patterns to the subtle arrangements of musical notes in Bachs Art of Fugue. So it is worth your while to spend some time (patiently, if necessary) to really understand how it works. As a byproduct, you will learn a valuable lesson about “complex numbers”.

If you already know all this, please skip down to the section on Mandelbrot, Dirac and Bach.

10.7.1 The iterative procedure defining Mandelbrot set

As we said, the definition is simplicity itself:

Take any number (x_{0}) and calculate the next one (x_{1}) and then the next one (x_{2}), then x_{3}, x_{4}, and so on, according to the prescription:

x_{n+1} = (x_{n})^{2} + x_{0}

for n= 0, 1, 2, 3, ... . In words: start with an initial number x_{0} - a “candidate for Mandelbrot membership”. Then determine the next number as “the previous number squared plus the initial number, and keep repeating this. This is known as a recursive formula” where there is a prescription how to calculate the next member of the series if you know the preceding one. This, and a choice for the very first member (x_{0}) allows sequential calculation of all the members of the sequence. We then say: if the sequence of numbers x_{n} escapes to infinity, then we say that the starting number x_{0} is not a member of the Mandelbrot set; otherwise x_{0} is a member of the set.

The recursive formula

x_{n+1} = (x_{n})^{2} + x_{0}

may look simple enough, but perhaps somewhat dense to people without much math background, so here are two examples to illustrate the process:

a) start with x_{0} = 1 and apply the recursive equation for n=0,1,2 ...:

x_{1} = x_{0}^{2} + x_{0} = 1^{2} + 1 = 1 + 1 = 2
x_{2} = x_{1}^{2} + x_{0} = 2^{2} + 1 = 4 + 1 = 5
x_{3} = x_{2}^{2} + x_{0} = 5^{2} + 1 = 25 + 1 = 26

and you dont need to be a Math major to see that this goes, inexorably, to infinity. Therefore, the number +1 is not a member of the Mandelbrot set.

b) now take a different starting number: x_{0} = −1 and calculate:

x_{1} = x_{0}^{2} + x_{0} = (−1)^{2} + (−1) = 1 − 1 = 0
x_{2} = x_{1}^{2} + x_{0} = 0^{2} + (−1) = 0 − 1 = −1
x_{3} = x_{2}^{2} + x_{0} = 0

Clearly, x_{4} = −1, x_{5} = 0 and so on: the numbers alternate between 0 and -1, so the sequence never escapes to infinity, and the number −1 is a member of the set.

So this procedure really is simple, but the resulting set is quite simple, too - in fact it is boring: with just a little more math one can show that the set is just a straight line starting at x_{0} = −2 and ending at x_{0} = +0.25.

10.7.2 Complex numbers

Things become interesting indeed when we keep the same procedure as above, but instead of “ordinary” numbers we allow the numbers to be ”complex”. The origin of the general need for complex numbers is the desire to be able to reverse the operation of squaring: we know that 2^{2} = 4, 3^{2} = 9 and so on. The reverse is called a ”square root”: the number whose square is 25 is called ”square root of 25”, and it is equal 5, because 5^{2} = 25. Similarly, the two statements

7^{2} = 49

and

\sqrt{49} = 7

are equivalent. The key to complex numbers is to ask yourself a question: what is the square root of −1, i.e. what is the number x such that x^{2} = −1. Obviously, no ordinary number x has the property that x^{2} = −1. But this does not stop our imagination: we imagine a new category of numbers, call them “imaginary numbers, and define one of them as i = \sqrt{−1} i.e. i^{2} = −1. It turns out that all other imaginary numbers are just plain multiples of i, for example:

\sqrt{−9} = \sqrt{9.(−1)} = \sqrt{9}\sqrt{−1} = 3i

and so on. And finally, by adding an ordinary (or real) number x and an imaginary number iy, we get a complex number z

z = x + iy

so that d = 1 + 2i, e = 1.5 − .5i, f = 2.5 and g = −1.2i are examples of complex numbers. And finally, we plot complex numbers as points in a plane, with the real part in the xdirection (horizontal) and the imaginary part in the y-direction (vertical)- see Figure 1.5.

10.7.3 Complex numbers are very powerful

We will analyze the driven damped oscillator in two ways: furst without complex numbers. Start with the equation of motion, i.e the 2nd Newton law:

ma(t) = F (t) = −x(t)k − Rv(t) + D sin(ωt)  (10.1)

where m is the mass, F(t) is the total force acting on the mass m, k is the stiffness of the spring, R is the damping coefficient, D is the amplitude of the driving force and ω = 2πf is the angular frequency of the driving force. We know that the steady state motion must be oscillatory with the angular frequency ω so we seek a solution in the form

x(t) = A sin(ωt + ϕ)

and we need to solve the equation of motion for the unknown amplitude of oscillation A and phase ϕ.

With our understanding of the rates of change, we know that

v(t) = \frac{dx(t)}{dt} = \frac{d}{dt}A sin(ωt + ϕ) = Aω cos(ωt + ϕ)

a(t) = \frac{d^{2}}{dt^{2}} x(t) = \frac{dv(t)}{dt} = \frac{d}{dt}Aω cos(ωt + ϕ) = −Aω^{2} sin(ωt + ϕ)

Plugging these expressions into the equation of motion 10.7.3 and using the trigonometric formulas

sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β − sin α sin β

we get

−mAω^{2} sin ωt cos ϕ − mAω^{2} cos ωt sin ϕ =
= −Ak cos ωt sin ϕ − Ak sin ωt cos ϕ −
− RAω cos ωt cos ϕ + RAω sin ωt sin ϕ +
+ D sin ωt

Now we collect all terms containing sin ωt on the LHS, and the terms containing cos ωt on the RHS:

sin ωt [− mAω^{2} cos ϕ+Ak cos ϕ−RAω sin ϕ−D] = cos ωt [ mAω^{2} sin ϕ−Ak sin ϕ−RAω cos ϕ] The expressions in square brackets are constants (i.e. independent of t) so that the equation has the form

C_{1} sin ωt = C_{2} cos ωt

and the great insight (used in various forms all over math and physics) is that the only solution of this equation can be the obvious and trivial choice C_{1} = 0 and C_{2} = 0, i.e.:

C_{1} = − mAω^{2} cos ϕ + Ak cos ϕ − RAω sin ϕ − D = 0  (10.2)

C_{2} = Amω^{2} sin ϕ − Ak sin ϕ − ARω cos ϕ = 0  (10.3)

Equation 10.3 immediately yields the solution for the phase:

tan ϕ = \frac{−R}{−mω + \frac{k}{ω}}  (10.4)

Now we can use the fact that if tan ϕ = a/b and defining s = \sqrt{a^{2} + b^{2}} we get^{7}

sin ϕ = \frac{a}{s} cos ϕ = \frac{b}{s}

Equation 10.2 then becomes

A\frac{−mω^{2}b + kb − Rωa}{s} = A\frac{(−mω^{2} + k)b − ωRa}{s} = Aω(b^{2} + a^{2})/s = Aωs^{2}/s = Aωs = D

and putting back the expressions for a, b and s gives the final result for the amplitude A

A = \frac{D}{ω\sqrt{R^{2} + (\frac{k}{ω} − mω)^{2}}}  (10.5)

And yes, this derivation is a little long^{8} but very beautiful and educational, and it is remarkable that so much can be done with just the sine and cosine and their rates of change.

Complex simplification:

We now formulate the Newton’s law with a complex displacement x(t) = Ae^{i(ωt+ϕ)} and driving force d(t) = De^{iωt}. Then the complex velocity and acceleration are

v(t) = dx(t)/dt = iωe^{i(ωt+ϕ)} a(t) = dv(t)/dt = (iω)^{2}e^{i(ωt+ϕ)}

and the Newton equation (equation ) is

7Putting an additional negative sign in front of both a and b does not change the value of tan ϕ, however it does influence equation 10.2. In our case, that solution would imply that the amplitudes A and D have opposite sign which is impossible (both are magnitudes, so both must be positive). It is also unphysical, as it would result in the displacement leading the force, instead of lagging (see Feynman).

8The derivation can be shortened by various tricks; it is presented here in the simplest possible way. But when we expand our treatment of complex numbers in PaM, we will see a really powerful simplification.

ma(t) = F_{total}(t) = mAe^{iϕ}(iω)^{2}e^{iωt} = −Ae^{iϕ}ke^{iωt} − Ae^{iϕ}R(iω)e^{iωt} + De^{iωt}

Canceling the common factor e^{iω} and re-arranging the terms we get

Ae^{iϕ} = \frac{D}{k − mω^{2} + iRω} = \frac{D}{k − mω^{2} + iRω} \frac{k − mω^{2} − iRω}{k − mω^{2} − iRω} = \frac{D(k − mω^{2} − iRω)}{(k − mω^{2})^{2} + R^{2}ω^{2}}

from which we immediately read off the results:

A = \frac{D}{\sqrt{(k − mω^{2})^{2} + R^{2}ω^{2}}}

tan ϕ = \frac{−Rω}{k − mω^{2}}

Compare this derivation with the one without complex numbers. Complex numbers are powerful (and beautiful) indeed!

But even then, the power and beauty of this pales compared to the role of complex numbers in Quantum Mechanics, and in the complex (”contour”) integration.

We are now ready to extend the Mandelbrot set to the complex plane. We simply repeat the basic procedure above, except now we are free to choose for the initial value an arbitrary complex number z_{0}, corresponding to a pixel in the display being prepared. We then evaluate^{9} the sequence:

z_{n+1} = (z_{n})^{2} + z_{0}

for n= 0, 1, 2, 3, ... . If the sequence of complex numbers z_{n} escapes to infinity, then we say that the starting complex number z_{0} is not a member of the Mandelbrot set; otherwise z_{0} is a member of the set. If z_{0} indeed turns out to be a member, make he pixel black, otherwise we leave it blank. In any case, we repeat this procedure for all pixels of our display, and we are done.

It should be obvious by now that the Mandelbrot set will be some kind of a blob in the x-y plane, located close to (x=0, y=0) - large initial numbers will get even larger when squared, and will quickly escape to infinity. And if asked for a prediction as to the shape of that blob, a reasonable guess might be: a circular or an elliptical disk or some other simple shape.

10.7.4 Mandelbrot, Dirac and Bach

However, the shape of the set is anything but simple. In spite of the simplicity of the algorithm, calculations to determine if an arbitrary complex number is or is not a member of the set are time consuming, and it took modern computers to even discover that the set looks interesting. Today, many free software packages for exploration of the Mandelbrot set are available on www - but be warned: this is addictive! Many of these programs use color, to indicate how close a given point is to the set. I find the use of color confusing - a point in the complex plane either is a member, or is not - and therefore I use black-and-white printing, with a procedure of my design to ensure that the thin filaments are not missed. Figure 1.6 shows the whole set - it is a beetle-like object, with cardioid-shaped main body, and many protuberances. It is upon zooming at various regions that you discover the true fractal nature of the set: the complexity of the set is inexhaustible. Delicate webs of filaments contain small, distorted versions of the original ”beetle” - and by zooming on the periphery of that beetle, you find delicate webs of filaments containing small, distorted beetles ... and so on and so on. As you travel along the beetle surface, the webs are self-similar but not identical, and they smoothly morph from baroque ornamentations to jagged snowflakes and exuberant spirals ( see Fig. 1.7). The extraordinary variety of the way in which the beetles are nestled in the filaments is illustrated on Fig. 1.8. The highest zoom I generated (the neighborhood of the point

z_{0} = −.7541521031327 + 0.050815525271 i

magnified by a factor of over 100,000,000 - see Figure 4 top left) shows the distorted beetle surrounded by filaments which are so dense that the resolution of my printer - and of my eye as well - is not sufficient to see the details.

9The Mandelbrot Appendix contains more explanations and technical details about the set and about complex numbers in general

People often use to artificially add color to displays of the Mandelbrot set by calculating the isoclines on the ”Mandelbrot mountain” and coloring the different height bands. This leads to sometime psychodelic, sometime beautiful (and sometimes both) images - see Figures 1.9 – 1.12 for some spectacular examples. This is one of the lovely cases where science meets art in a happy cooperation.

Mandelbrot set is an infinitely complex Universe generated by an absurdly simple Law. It should remind you of those nerdish Physics students with Maxwell equations on their Tshirts. As the great Dirac said, the simple equations of electromagnetic interactions between the nuclei and electrons(in their quantized form) govern ”most of Physics and all of Chemistry” - and he considered the ”problem of life” as just another, just a little more difficult Physics[sic] problem ... On the other side of the exuberance - humility spectrum, I am reminded of the Mandelbrot set by the structure of Bach’s fugues, especially in the Art of Fugue. As I argue in the detailed description of many of the fugues, the main Theme and its variations can be seen as the Mandelbrot ”beetles”, all the way to the distortions upon ”zooming”. And the various motifs which accompany the Theme – and which make the fugues so interesting – cannot but remind me of the delicate filaments.

Finally note that the Mandelbrot set only became interesting when we switched from real numbers to complex numbers. This alone would have made the invention of complex numbers worthwhile. But in fact, complex numbers provide powerful and beautiful tools in innumerable Math, science and engineering applications, and we shall see that quantum mechanics (and therefore the very existence of the Universe) would be impossible without complex numbers. I believe that a brief introduction such as the one in this chapter should be part of the standard high school curriculum.

10.7.5 Vladibrot Set

Note that a different, apparently equally simple procedure does lead to a simple set. To illustrate this, I propose, as a very useful exercise, to consider a “Vladibrot Set”^{10} defined by a procedure:

start with a complex number z_{0} = x_{0} + iy_{0}. Then keep evaluating

z_{n+1} = z_{n}\sqrt{x_{n}^{2} + y_{n}^{2}}

for n= 0, 1, 2, 3, ... . If the sequence of complex numbers z_{n} escapes to infinity, then we say that the starting complex number z_{0} is not a member of the Vladibrot set; otherwise z_{0} is a member of the set.

This procedure looks as complicated (if not more) as the Mandelbrot procedure; yet the shape of the resulting Vladibrot set is simple indeed - determine the shape as an exercise. Hint: from Dr. Pythagoras we know that \sqrt{x^{2} + y^{2}} is simply the “magnitude” of the complex number z = x + iy - see Figure 1. Therefore, for the Vladibrot procedure, all the numbers z_{n} are “in the same direction” as the original z_{0} - they just have different magnitudes (“lengths”). The solution of this exercise can be found at the book website. However, I urge the reader not to give up easily. You will see why the Vladibrot set never became famous ...

10Last year, a student wondered how I could have the chutzpah to name a set after myself. Please see the last sentence of this section and construct the “Vladibrot” set to appreciate my self-deprecating joke.

10.7.6 Back to the complex Mandelbrot

Consider first the examples of complex numbers given in the main text: d = 1 + 2i, e = 1.5 − .5i, f = 2.5 and g = −1.2i The real and the imaginary parts of the above complex numbers are d=(1.,2.) e=(1.5,-.5) f=(2.5,0.) and g=(0.,-1.2) respectively. Note that the real numbers (e.g. the f = 2.5 above) and imaginary numbers (e.g. g = −1.2i above) are special cases of complex numbers.

We need to be able to add and multiply complex numbers. Adding is easy: add the real parts to obtain the real part of the sum, add the imaginary parts to get the imaginary part. So for example:

d + e = (1 + 2i) + (1.5 − .5i) = (1 + 1.5) + i(2 − .5) = 2.5 + 1.5i

Multiplication is not much more difficult: just remember that i^{2} = −1:

d × e = (1 + 2i)(1.5 − .5i) = 1 × 1.5 + 1 × (−.5i) + 2i × (1.5) + 2i × (−.5i) =
= 1.5 − .5i + 3i − 1 × i^{2} = 1.5 + 1 + i(−.5 + 3) = 2.5 + 2.5i

Let us test^{11} a purely imaginary starting point z_{0} = i for Mandelbrot membership

z_{1} = z_{0}^{2} + z_{0} = i^{2} + i = −1 + i
z_{2} = z_{1}^{2} + z_{0} = (−1 + i)^{2} + i = ((−1)^{2} + 2(−1)i + i^{2}) + i = (1 − 2i − 1) + i = −i
z_{3} = z_{2}^{2} + z_{0} = (−i)^{2} + i = −1 + i

In this case, we are lucky: z_{3} turns out to be is the same as z_{1} - therefore, the sequence will oscillate between −i and −1 + i, never escaping to infinity. Therefore, z_{0} = i is a member of the Mandelbrot set.

Together with our previous results from the real axis, we have determined two members of the Mandelbrot set: z_{0} = −1 and z_{0} = i (see Figure 1). As an exercise, please verify that z_{0} = −2 and z_{0} = −i are members of the set, and z_{0} = −2i is not.

11note: to calculate (−1 + i)^{2} I used the fact that (a + b)^{2} = (a + b)(a + b) = a^{2} + 2ab + b^{2} - and of course i^{2} = −1

136 CHAPTER 10. PHYSICS AND MATH APPENDIX

10.8 Figures

10.8. FIGURES 137 Figure 10.1: Graphical representation of numbers in the “complex plane”.

Figure 10.2: Feynman’s Jewel: e^{iΦ} = cos Φ + i sin Φ

First check that the procedure works for e^{x} with a real exponent: the first three columns show the values of x (A), the results of the iterative procedure as calculated (B), compared with the values of e^{x} from my calculator (C). The procedure does work. Showing more decimal places would show small differences (that would be cured by dividing x into even smaller pieces, and so on).

Now apply the procedure on e^{iΦ}. The first two columns are: the angle Φ in Radians (E) and degrees (F) next 2 columns: Imaginary part of e^{iΦ} as calculated (G), compared to sin Φ from the calculator (H); next 2 columns: Real part of e^{iΦ} as calculated (I) and compared to cos Φ (J)

So we see that indeed e^{iΦ} = cos Φ + i sin Φ

Technical “detail”: Following Feynman’s clever way of dealing with the rounding errors, we first proceed by doubling the argument (the first 14 rows) then switch to increasing the argument linearly (by 0.1024 Radians for e^{x} and by 15 degrees for e^{iΦ}).

Figure 10.3: Top: (Wolfram) Mathematica plot of the function f_{0}(x) = e^{xSqrt(Sign(x))} blue=Re(f_{0}), yellow = Im(f_{0}) and green is the magnitude \sqrt{Re^{2} + Im^{2}} Bottom: The above function f_{0}(x) correctly depicts the sine and cosine for left-hand (imaginary) side adding to a constant magnitude, and the purely real exponential on the right-hand side. It is continuous at x=0 but the derivative is not – after all, it corresponds to an abrupt change of direction from descending along the real axis to increasing along the perpendicular imaginary axis. Also, it is not a “genuine” function but a “trick” to combine two functions together. One elegant way to improve the illustration is to plot the function f_{5}(x) = e^{x^{5/2}} shown here - the transition between oscillatory and exponential behavior is smooth.

I cannot resist to mention that when I first produced this picture, I was somewhat reminded of a Bach fugue motive (this may be a variant of the Rohrschach test joke – I just see Bach everywhere ....)

140 CHAPTER 10. PHYSICS AND MATH APPENDIX Figure 10.4: Caption for deltaAnalytic Figure 10.5: The complex plane z = x + iy with several complex numbers indicated as examples. The line segment from x=-2 to x=+0.25 is the ”Mandelbrot set on the real axis”. The sketch shows the rough outline of the Mandelbrot set proper (i.e. in the complex plane)

- see Fig. 1.2 for the real thing.

Figure 10.6: The Mandelbrot set. Three small secondary beetles are visible ”by the naked eye” (one at z_{0} = −1.75, the other two slightly outside of y = ±i). Zooming reveals an infinite number of such beetles, each similar but not identical to the main one. The appearance of the filaments which connect the beetles depends on the regions on which you zoom (see Fig. 3 and 4). This sets the Mandelbrot set apart from other fractal sets, which tend to be self-similar upon zooming, in a somewhat boring way.

10.8. FIGURES 143 Figure 10.7: Two relatively low-level zooms showing details of the periphery of the main beetle.

Figure 10.8: Four deep zooms showing different ways in which the ”beetles” are nestled in the elaborate webs of filaments. The complexity of the web at the top left (at a zoom factor of more than 100,000,000) is such that the software resolution is not sufficient to show the details.

10.8. FIGURES 145 Figure 10.9: Example of a deep zoom into the set, using colors to indicate the height of the Mandelbrot ”mountain” at various isoclines.

146 CHAPTER 10. PHYSICS AND MATH APPENDIX Figure 10.10: Another color example. Note the small beetle embedded in the pattern at the bottom center.

Figure 10.11: Zoom at the small beetle on previous Figure. Zooming on its periphery will provide additional detail, reminiscent to what we already saw before, but never the same. And this goes on for ever!

148 CHAPTER 10. PHYSICS AND MATH APPENDIX Figure 10.12: One more color example, of exquisite beauty, with a miniature beetle calling for additional axploration.