This page lists (before the lectures) what is planned to be was covered, and then (after the lectures) what was actually covered. It also lists material that you are expected to read that was not covered, or only partially covered, in lectures. Handouts are also archived here.
To bottom (most recent lectures)| Lecture | Date | Covered in lecture | Reading (not covered or partially covered) |
| 1.1 | 9/30 |
Class organization. Sakurai 1.1: Stern-Gerlach experiment and spin in QM. ( Notes ) |
Analogy with circular polarization in Sak. 1.1. |
| 1.2 | 10/1 |
This will be a regular lecture. Recap on complex vector spaces. Dirac notation (Sak. 1.2). First handout . |
|
| 2.1 | 10/5 |
More quantum kinematics, mainly from Sakurai sections 1.2 and 1.3: Spin bases, operators, outer products, hermitian operators, orthonomal bases. ( Notes ) |
You will have to read ahead to the end of 1.3 to cover all the material on HW1 |
| 2.2 | 10/7 |
Matrix representations of operators, trace, determinant. Measurement theory in QM (Sakurai 1.4) ( Notes ) |
What happens when eigenvalue measured is degenerate? What replaces the projection operator in this case? Read last page of notes (4.10) |
| 2.3 | 10/8 |
Wrap up discussion of measurement Compatible and incompatible observables ( Notes ) |
|
| 3.1 | 10/12 |
Heisenberg Uncertainty Principle (Sak. 1.4) Unitary operators and basis changes (Sak. 1.5) ( Notes ) |
Read "Unitarily equivalent observables" (Sak. 1.5) Do exercise given in class. |
| 3.2 | 10/14 |
Continuously labeled basis kets (non-normalizable) Classical to quantum "transition" for a single, spinless particle (Dirac's correspondence, begin Sakurai's approch) ( Notes ) |
Read 48-51 in Sakurai (last four pages of section 1.6) |
| 3.3 | 10/16 |
Finish Sakurai's discussion of classical to quantum transition Momentum, position and translation ops (Sak 1.6) Momentum operator in position basis, momentum basis states (Sak 1.7) ( Notes ) |
|
| 4.1 | 10/19 |
Finish Sakurai 1.7: transforming between position and
momentum bases and Gaussian wave packets Begin Ch. 2: time evolution operator, Schrodinger equation, general solutions. ( Notes ) |
Read last part of Sakurai Ch. 1 concerning wave packets. |
| 4.2 | 10/21 |
Example of time evolution: precession of a quantum spin Schrodinger vs. Heisenberg pictures ( Notes ) |
Read Sakurai pages 78-80: Energy-time "uncertainty" relation. |
| 4.3 | 10/22 |
Examples of Heisenberg equations of motion: Free particle, Ehrenfest theorem, spin-1/2 precession ( Notes ) |
|
| 5.1 | 10/26 |
Wave-packet spread Time evolution backward in time Heisenberg picture base-kets (Sak. 2.2, pp87-89) Begin Simple Harmonic Oscillator (Sak. 2.3) First pretest . ( Notes ) |
|
| 5.2 | 10/28 |
General considerations for solving time indep. Schrodinger eq: graphical analysis, asymptotic forms, reality of wavefunctions, discrete and continuous spectra. Parity operator ( Notes ) |
I am assuming that you have some familiarity
with the derivation of SHO wavefunctions in terms of
Hermite polynomials. Please read
Sakurai A.4 to refresh your memory.
The other topics are scattered in Sakurai, or appear in any other QM text. |
| 5.3 | 10/29 |
Finish discussion of parity operator Dirac's operator solution of SHO ( Notes ) |
|
| 6.1 | 11/2 |
Matrix elements in the SHO Variational priniciple and application to SHO Coherent states in the SHO ( Notes ) |
For more on coherent states see Gottfried and Yan,
pp181-4. |
| 6.2 | 11/4 |
Connecting our definition of coherent states
with the standard one Squeezed states in the SHO Some odds and ends on the SHO ( Notes ) |
For more on squeezed states, you'll need to look beyond the standard texts. Wikipedia has some good references. |
| 6.3 | 11/5 | Review for midterm | |
| 7.1 | 11/9 |
MIDTERM Solution is here . |
|
| 7.2 | 11/12 |
1-d scattering: quick review Solving an example scattering problem ( Notes ) |
You should be able to reproduced the results
collected in Sakurai appendices A.1-A.3. Sakurai has little discussion of 1-d scattering, but most quantum texts do. See for example GY sec 4.4. |
| 8.1 | 11/16 |
Plots of transmission probability for potential barrier, and
Mathematica notebook to create them. Interpretation of time-independent scattering solutions using probability current density Begin discussion of scattering using wave packets ( Notes ) |
Probability current density is discussed on Sakurai pp101-3.
See Baym for a nice discussion of scattering using wave packets. |
| 8.2 | 11/18 |
Complete discussion of scattering using wave packets Resonances Various plots shown are here (or in Mathematica notebook ). Begin discussion of bound-state energies from poles in T and R. ( Notes ) |
|
| 8.3 | 11/19 |
Complete discussion of bound states Begin discussion of WKB approximation: derivation. ( Notes ) |
Read Sakurai pp.103-109. |
| 9.1 | 11/25 | Classes canceled! | |
| 9.2 | 11/25 | HOLIDAY! | |
| 9.3 | 11/26 | HOLIDAY! | |
| 10.1 | 11/30 |
WKB approximation in 1-d: general solution, matching equations,
application to bound state problems Handout and notes . Mathematica notebook used to obtain WKB and exact wavefunctions for example of SHO with hard walls and quartic potential. |
For further reading, see Gottfried and Yan, sec 4.5(b),
and L.S. Brown, Am. J. Phys. 40, 371 (1972)
["Classical limit and the WKB approximation"]
and 41, 526 (1973)
["Classical limit of the Hydrogen Atom"]. Notes and handout for an additional lecture (not given due to snow day) on the decay of resonances using wave-packets and the WKB approximation. This is adapted from B.R. Holstein, Am. J. Phys. 64, 1061 (1996) "Understanding alpha decay". |
| 10.2 | 12/2 |
Derivation of
path integral representation of transformation
function. Some brief comments on propagators. Notes . |
Read Sakurai p109-116 on propagators. I will not cover all
this in lectures but expect you to have read it.
For further reading on path integrals see not only Sakurai but also Gottfried & Yan and the classic, Feynman & Hibbs ("Quantum Mechanics and Path Integrals", McGraw-Hill, 1965). Also good is the first chapter of Lowell S. Brown, "Quantum Field Theory" (Cambridge, 1992). |
| 10.3 | 12/3 |
Calculating the SHO transformation function using
path integrals. Notes . |
Feynman and Hibbs. |
| 11.1 | 12/7 |
E and B fields in classical mechanics. E and B fields in QM---Heisenberg equations of motion, Landau levels, probability current. Begin gauge transformations in QM. Notes . |
Read Sakurai, section 2.6, which gives a nice discussion. We will not cover all the material in this section, but you will delve into Landau levels more deeply in HW8. |
| 11.2 | 12/9 |
More on gauge transformations in QM. Aharanov-Bohm effect (Refs for experiments: R.G. Chambers, Phys. Rev. Lett. 5 (1960) 3 and A. Tonomura et al, Phys. Rev. Lett. 48 (1982) 1443.) Notes . |
Read in Sakurai 2.6 about
gravitationally induced quantum interference (Experimental refs: R. Colella et al, Phys. Rev. Lett. 34 (1975) 1472 and H. Kaiser et al., Physica B385 (2006) 1384.) |
| 11.3 | 12/10 | Review for final exam | |
| 12 | 12/15 |
FINAL EXAM (10:30-12:20)
Solution is
here . |