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 | 10/1 |
Class organization. Sakurai 1.1: Stern-Gerlach experiment and spin in QM. |
Analogy with circular polarization in Sak. 1.1. |
| 1.2 | 10/2 |
This will be a regular lecture. Recap on complex vector spaces, including Hilbert space. Dirac notation (Sak. 1.2). First handout . |
|
| 2.1 | 10/6 |
More quantum kinematics, mainly from Sakurai sections 1.2 and 1.3: Spin bases, operators, hermitian operators, orthonomal basis |
You will have to read ahead to the end of 1.3 to cover all the material on HW1 |
| 2.2 | 10/8 |
Matrix representations of operators, trace, determinant. Measurement theory in QM (Sakurai 1.4) |
|
| 2.3 | 10/9 | Compatible and incompatible observables | |
| 3.1 | 10/13 |
Heisenberg Uncertainty Principle Unitary operators and basis changes |
|
| 3.2 | 10/15 |
Quantum to classical transition Momentum, position and translation ops (Sak 1.6) |
Read 48-51 in Sakurai (last four pages of section 1.6) |
| 3.3 | 10/16 | Sakurai 1.7: Momentum operator in position basis, momentum basis states. | |
| 4.1 | 10/20 |
Finish Sakurai 1.7: transforming between position and
momentum bases and Gaussian wave packets Begin Ch. 2: time evolution operator U, Schrodinger equation for U, some general solutions. |
Read last part of Sakurai Ch. 1 concerning wave packets. |
| 4.2 | 10/22 |
Example of time evolution: precession of a quantum spin Schrodinger vs. Heisenberg pictures |
Read Sakurai pages 78-80: Energy-time "uncertainty" relation. |
| 4.3 | 10/23 |
Ehrenfest theorem Spin precession using Heisenberg equations of motion |
|
| 5.1 | 10/27 |
Wave-packet spread Heisenberg picture base-kets Begin Simple Harmonic Oscillator First pretest . |
|
| 5.2 | 10/29 |
Discussion of pretest: General considerations for solving time indep. Schrodinger eq. Heisenberg eqs. of mtn. for SHO Using the variational principle |
I am assuming that you have seen in UG QM the derivation of SHO wavefunctions in terms of Hermite polynomial, i.e. of the results of Sakurai A.4 |
| 5.3 | 10/30 | Review for midterm. | |
| 6.1 | 11/3 |
MIDTERM Solution is here . |
|
| 6.2 | 11/5 |
Variational estimate of SHO ground state energy. Parity operator in context of SHO Solving SHO using Dirac's operator method |
Various calculations are left as exercises on HW5. |
| 6.3 | 11/6 |
Discussion of midterm results Wrapping up solution of SHO Dynamics in SHO |
|
| 7.1 | 11/10 | Coherent and squeezed states in the SHO |
For more on coherent states see Gottfried and Yan,
pp181-4. For more on squeezed states, you'll need to look beyond the standard texts. Wikipedia has some good references. |
| 7.2 | 11/12 |
Pretest 2 1-d scattering: quick review Solving an example scattering problem Plots of transmission probability, and Mathematica notebook to create them. Probability current. |
You should be able to reproduced the results
collected in Sakurai appendices A.1-A.3. See Baym for a nice discussion of scattering using wave packets. Probability current density is discussed on Sakura pp101-3. |
| 7.3 | 11/13 |
Interpretation of time-independent scattering solutions
using probability current density Begin discussion of scattering using wave packets |
|
| 8.1 | 11/17 |
Complete discussion of scattering using wave packets Resonances Bound states from poles in transmission amplitude Various plots shown are here (or in Mathematica notebook ). |
|
| 8.2 | 11/19 |
WKB approximation: derivation and one application Plots of wavefunction for example of SHO with hard walls, both WKB and exact, and Mathematica notebook to create them (and plots for next lecture). |
Read Sakurai pp.103-109. |
| 8.3 | 11/20 |
Connecting WKB approximation across classical turning points Example of quartic potential Plots comparing exact and WKB wavefunctions. |
For further reading, see Gottfried and Yan, sec 4.5(b) Plots comparing exact, WKB and Airy functions. (Obtained using updated notebook posted above). Further reading on WKB: 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"]. |
| 9.1 | 11/24 |
Decay of resonances (e.g. alpha decay): an
application of wave-packets and the WKB approx. Handout , figures and notebook used in class. |
This is adapted from B.R. Holstein, Am. J. Phys. 64, 1061 (1996) "Understanding alpha decay" |
| 9.2 | 11/26 | HOLIDAY! | |
| 9.3 | 11/27 | HOLIDAY! | |
| 10.1 | 12/1 |
Propagators (briefly) and derivation of
path integral representation of transformation
function. For further reading see not only Sakurai but also Gottfried & Yan and the classic, Feynman & Hibbs ("Quantum Mechanics and Path Integrals", McGraw-Hill, 1965). |
Read Sakurai p109-116 on propagators. I will not cover all this in lectures but expect you to have read it. |
| 10.2 | 12/3 |
Calculating the SHO transformation function using
path integrals. E and B fields in classical mechanics. |
|
| 10.3 | 12/4 | HW review (run by Sichun Sun). | |
| 11.1 | 12/8 |
E and B fields in QM---Heisenberg equations of motion,
probability current, Landau levels. Gauge transformations in QM. |
Read Sakurai, section 2.6. |
| 11.2 | 12/10 |
More on gauge transformations in QM. Aharanov-Bohm effect (Refs: R.G. Chambers, Phys. Rev. Lett. 5 (1960) 3 and A. Tonomura et al, Phys. Rev. Lett. 48 (1982) 1443.) Gravitationally induced quantum interference (Refs: R. Colella et al, Phys. Rev. Lett. 34 (1975) 1472 and H. Kaiser et al., Physica B385 (2006) 1384.) |
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| 11.3 | 12/11 | Review for final exam | |
| 12 | 12/16 | FINAL EXAM (10:30-12:20) |