Further Quantum Mechanics

9 lectures in HT and 12 lectures in TT

Prof J.J. Binney

This is the second of two courses that cover paper A3 in Part A of the three- and four-year physics courses. The syllabus is given in the Course Handbook.

Goals of the course

Exact solutions to equations carry you only so far in physics: to cover the richness of the real world, we have to extend exact solutions by approximate methods. This course shows how we do this.

The key to a time-independent system is solving its TISE. We extend exact solutions of the TISE to systems that have similar Hamiltonians. We also show how to obtain useful limits on the ground-state energy without solving the TISE, even for a similar system.

When the Hamiltonian is time-dependent, the TISE doesn't provide a simple solution to the TDSE, but we are able to obtain approximate solutions to the TDSE when the time dependence is weak and we can solve the TISE when the time dependence is neglected. These approximate solutions explain why quantum systems appear to jump between energy levels.

Quantum mechechanics introduces a bizarre new symmetry - exchange symmetry - between particles of the same species. We understand how exchange symmetry profoundly affects atoms with two electrons.

If the Hamiltonian changes slowly enough and we can solve the TISE for each fixed time, we can solve the TDSE for the real, time-dependent case. This trick opens up a wide field of physical applications.

In the classical world probability arises from ignorance, whereas the violation of Bell's inequality by QM killed Einstein's dream that better information would one day banish probability from the quantum world. But ignorance is universal and it broadens probability distributions even in the quantum world. We have to extend the formalism of QM to include ignorance. When we do, we see that ignorance must at the very least contribute to the probabilistic nature of Born's rule.

Synopsis

The presentation will closely follow that in The Physics of Quantum Mechanics, James Binney and David Skinner, Oxford University Press 2014, so no lecture notes will be issued. A slightly obsolete version is downloadable gratis at B&S. The division into lectures is only approximate

1 Time-independent perturbation theory (B&S Sec 10.1)

2 The quadratic stark effect (B&S Sec 10.1.1)

3 Degenerate time-independent perturbation theory (B&S Sec 10.1.2)

4 The linear Stark effect (B&S Sec 10.1.2)

5 Motion in a magnetic field (B&S Secs 9.1, 9.2)

6 The Pachen-Back effect (B&S Sec 10.1.3)

7 The variatonal principle (B&S Sec 10.2)

8 Time-dependent perturbation theory B&S Sec 10.3)

9 Fermi golden rule (B&S Sec 10.3.1)

10 Radiative transitions (B&S 10.3.2)

11 Selection rules (B&S Sec 10.3.3)

12 Identical particles (B&S Sec 11.1)

13 Gross structure of helium I (B&S Sec 11.2)

14 Gross structure of helium II (B&S Sec 11.2)

15 Adiabatic principle (B&S Sec 12.1)

16 Condensed matter & the adiabatic principle (B&S Sec 12.4)

17 Molecular dynamics & the adiabatic principle (B&S Sec 12.5)

18 The WKBJ approximation (B&S Sec 12.6)

19 Classical uncertainty in QM: density operators (B&S Sec 6.3)

20 Density operators & measurements (B&S Sec 6.5)

Problem sets

HT1 HT2 - Easter Vacation TT (Problem 3.7 refers to Appendix K, which is not included in B&S but is here)

Books

The Physics of Quantum Mechanics, James Binney and David Skinner, Oxford University Press 2014

The course is based on chapters 9 - 12 of this book.

A Modern Approach to Quantum Mechanics, J.S. Townsend. McGraw Hill published a paperback edition but now it seems to be available only in hardback at £54 from University Science Books. This is my favourite undergraduate text for QM because it has a clear logical structure and doesn't waste time on history.

Modern Quantum Mechanics, J.J. Sakurai. Addison Wesley gives quite a similar perspective to that of the course. Only available in hardback at about £50

Quantum Physics, Stephen Gasiorowicz, Wiley £25. A good traditional text at the right level. However, I don't like the traditional approach.

The Feynman Lectures vol 3, R.P. Feynman, R.B. Leighton & M. Sands, Addison Wesley £32. Feynman was one of the great physicists of the last century and had unique insight into QM that comes across in this book. Well worth studying to see how Feynman approaches a problem. The trouble is, his approach generally isn't algorithmic but depends on his remarkable physical insight, so it isn't easy to emulate.

The Principles of Quantum Mechanics, P.A.M. Dirac, Oxford University Press £31.35. Dirac contributed as much to QM as anyone and this book is based on the lectures he gave in Cambridge for the Mathematical Tripos. His arguments can be so elegant you miss them, so close reading is essential. But the power and clarity with which he lays out the ideas and obtains results is awesome. Not for the mathematically challenged, but if you like mathematics, do give it a try.