Quantum Mechanics

12 lectures in MT and 15 lectures in HT

Prof J.J. Binney

This is the first 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

After working through this course you should understand how probabilities are obtained from quantum amplitudes, why they give rise to quantum interference, the concept of a complete set of amplitudes and how this defines a "quantum state". You should understand why each observable is associated with a Hermitian operator, and how the latter is used to calculate the expectation value of the observable. You should understand the connection between the momentum and angular-momentum operators and translations and rotations of states, and the connection between this aspect of operators and commutation relations. You should have understood the connection between symmetries and conservation laws.

You should be familiar with the energy, position and momentum representations. You will have used the position representation to understand the position/momentum uncertainty principle and used the energy representation to study the dynamics of both harmonic and anharmonic oscillators. You will have studied the energy levels of particles interacting with various one-dimensional potential wells and used the results to understand the workings of an ammonia maser.

You will have determined the eigenvalues of the total and orbital angular-momentum operators and understood the difference betwen the transformations that they generate. You will have understood how the kinetic energy of a particle can be broken down into radial and tangential contributions. You will have constructed both the Pauli matrices to handle problems involving spin-half particles, and the corresponding matrices for spin-one particles. You should understand the connection between a particle's spin and orientation. You should be familiar with the gross-structure spectrum and stationary states of hydrogen.

There will not be time to describe the history of quantum mechanics. This is a fascinating story but rarely well told - most books try to combine the history with a potted account of the science. Worthwhile are "Inward Bound" by A. Pais (OUP) and "Fundamentals of Modern Physics" by J.J. Eisberg (Wiley). Werner Heisenberg's collection of essays "Across the Frontiers" (originally "Schritte ueber Grenzen") gives a gripping first-hand perspective on the later half of the development of quantum mechanics (and of life in Germany during the first half of the 20th c).

Synopsis

The presentation will closely follow that in The Physics of Quantum Mechanics, James Binney and David Skinner, Oxford University Press 2014. A slightly obsolete version is downloadable gratis at B&S. The division into lectures is only approximate

1 Amplitudes, quantum states, the energy representation

2 Dirac notation & the energy representation

3 Operators and observables

4 The TDSE, the position representation

5 Particle dynamics

6 Wave functions of well-defined momentum and the uncertainty principle

7 Two slits revisited, extensions to three dimensions, virial theorem

8 Harmonic oscillator: the stationary states

9 Dynamics of oscillators

10 Transformations and observables: transformations of kets

11 Transformations and observables: transformations of operators

12 Symmetries & conservation laws, geometry and commutators

13 The square well

14 A pair of square wells, ammonia maser

15 Tunnelling and radioactive decay

16 Composite systems & entanglement

17 Einstein-Podolski-Rosen R experiment & Bell inequalities

18 Angular momentum; eigen values of J_z and J^2

19 Spectra of diatomic molecules

20 Orbital angular momentum, generation of circular translations

21 Eigenfunctions of L_z and L^2; radial/tangential decomposition of KE

22 Spin

23 Stern-Gerlach experiment and spin 1

24 Classical spin & the addition of angular momenta

25 Gross structure of hydrogen

26 The emissinn spectrum of hydrogen

27 Eigenfunctions of hydrogen

Videos of the lectures

Problem sets

MT1 MT2 Christmas Vacation HT1 HT2 HT3 Easter Vacation

all

Books

The Physics of Quantum Mechanics, James Binney and David Skinner, Oxford University Press 2014 about £25

The course is based on the first 8 chapters of this book, and the remaining chapters cover the rest of the second-year material and some more.

A slightly obsolete version can be downloaded gratis from here

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 was once 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.