Course outline for advanced quantum mechanics II

Taught by: Duncan O'Dell (Office: ABB 320. Email: dodell@mcmaster.ca with any enquiries)

In AQM II we will look at the application of quantum mechanics to three main areas: 1) scattering theory, 2) many-particle systems (second quantization), and 3) relativistic single particles. Towards the end of the course the student will have the opportunity to give a presentation on a topic of their choice in quantum mechanics.

Detailed Syllabus:

Part I: Scattering theory

1 Motivation.

2 Review of two-body scattering.

3 Review of motion in a centrally symmetric field, including partial waves, ingoing/outgoing waves, normalization, and phase shifts.

4 The typical scattering experiment, including asymptotic scattered wave function, scattering amplitude and cross-section.

5 Low-energy scattering, including s-wave scattering length, and resonance scattering.

6 Analytic properties of the scattering amplitude.

7 Green’s function approach, including eigenfunction expansion of the Green’s function, spectral representation of the Green’s function, Lippmann-Schwinger wave equation.

8 Perturbation theory, including the Born approximation, and the T-matrix.

9 Identical particle scattering, including direct and exchange terms.

Books: L.D. Landau & E.M. Lifshitz Quantum Mechanics (non-relativistic theory), R.H. Landau Quantum Mechanics II

Part II: Second quantisation

1 Motivation.

2 Many body wave mechanics (bosons).

3 Occupation number representation (bosons), including one- and two-body operators.

4 Annihilation and creation operators (bosons).

5 The fermion case.

6 Basis independent representation of second quantization: the field operators, including spinless and spinful cases.

7 The non-interacting Fermi gas, including the density operator and the single-particle density matrix.

8 The pair correlation function for the non-interacting Fermi gas and the pair correlation function for non-interacting bosons, including the Hanbury Brown Twiss experiment.

9 The interacting electron gas in a uniform positive background: jellium, including first order Hartree and exchange terms, the correlation energy, and higher order perturbation theory.

10 Mean-field approach: the Hartree-Fock approximation.

11 Hartee-Fock theory for extended systems.

12 Off-diagonal long-range order in the single-particle density matrix, including Bose-Einstein condensation and coherent states.

13 Weakly interacting Bose gas, including the Bogoliubov transformation.

14 The BCS theory (Bogoliubov transformation approach).

15 Superfluidity, including the Landau criterion, Goldstone bosons and quantized vortices.

Books: A.L. Fetter & W.D. Walecka Quantum theory of many-particle systems, G. Baym Lectures on Quantum Mechanics, C. Kittel Quantum theory of solids, M. Tinkham Introduction to superconductivity, L. Pitaevskii and S. Stringari Bose-Einstein condensation.

Part III: Relativistic quantum mechanics

A: The Klein-Gordon equation

1 Motivation.

2 Introduction to the Klein-Gordon (KG) equation, including discussion of Lorentz invariance.

3 Properties of the free-particle KG eqn, including negative energy solutions.

4 First order in time equations, including positive and negative energy solutions.

5 Qualitative features of one-dimensional motion for KG eqn with an electrostatic potential.

6 Plane wave incident upon electrostatic potential barrier, including the Klein paradox and pair production.

B: The Dirac equation

1 The Dirac equation.

2 The Dirac matrices and Dirac algebra.

3 Spin of the Dirac particle.

4 Dirac particle in an electromagnetic field, including the Pauli equation, magnetic moment and the g-factor.

5 Free particle plane wave solutions, including negative and positive energy solutions.

6 Some more comments on the gyromagnetic ratio and related topics, including the Darwin term and Zitterbewegung, and the Lamb shift.

7 Covariant form of the Dirac equation, including gauge covariant substitution for particle in electromagnetic field.

8 Lorentz invariance, including Chiral spinors, helicity and the neutrino.

9 Transformation properties of Dirac bilinear covariants.

10 Dirac lagrangian.

11 Plane waves and Lorentz boosts.

12 Do electrons travel at the speed of light?

13 The Dirac sea: negative energy solutions, hole theory and charge conjugation.

14 Hydrogen fine structure.

15 The road to quantum field theory (second quantization again).

Books: P.A.M. Dirac, The principles of quantum mechanics, G. Baym Lectures on Quantum Mechanics, J.D. Bjorken and S.D. Drell Relativistic Quantum Mechanics, B. Holstein Topics in Advanced Quantum Mechanics, L. Schiff Quantum Mechanics, P.W. Milonni The quantum vacuum.