Fundamental physics, year 1 (Teknisk fysik, årskurs 4)

Advanced quantum mechanics — Lecture notes

by Christian von Schultz

Lecturers: Gabriele Ferretti, Stellan Östlund and Bernhard Mehlig

The course home page can be found at http://bohm.fy.chalmers.se/~ostlund/kvant3/.

Tuesday 2008-10-28

New lecturer: Gabriele Ferretti. Scattering. Cross section σ. Flux. Event. Differential cross section. The Lippmann-Schwinger equation. Scattering from potential ↔ elastic scattering of two particles. Perturbation theory in a continuum. Lippmann-Schwinger in the coordinate representation. The residue theorem.

Wednesday 2008-10-29

More on the Lippmann-Schwinger equation. What goes wrong if we choose -iε in the denominator. Incoming and outgoing radial waves. The current associated with a wave function. The scattering amplitude f(p, p'). The relation between scattering amplitude and differential cross section. The first order Born approximation. Going from the centre of mass system to the laboratory frame (hmm, we did not seem to finish that).

Friday 2008-10-31

I did not take notes this day. Sorry. I had to visit the hospital.

Tuesday 2008-11-04

The eikonal approximation. Partial waves.

Wednesday 2008-11-05

Partial waves. Partial wave scattering amplitude. Argand diagrams. Connection eikonal — partial wave. Hard sphere s-wave scattering. Analytic properties of the S matrix. Resonances.

Friday 2008-11-07

Scattering with identical particles. Parity and time reversal. The detailed balance. Inelastic scattering. The size of a nucleus. Nuclear form factor. The Hilbert space, creation and annihilation in quantum field theory.

Tuesday 2008-11-11

We only discussed the take home exam. In problem two, averaging over the direction of the incoming beam is good.

Wednesday 2008-11-12

New lecturer: Stellan Östlund. Multi-particle systems. Kets with angle brackets and with round parenthesis. Occupation number of the states. ξ = 1 for bosons, ξ = -1 for fermions. Distinguishable and indistinguishable particles. The Slater determinant. For bosons: the permanent. The particle number representation. The Fock-space. Creation and annihilation operators.

Friday 2008-11-14

Bosons and fermions: commutation and anti-commutation relations. Matrix representation of creation operators. Unitary transformation. Commutation with the number operator. The Hamiltonian. Second quantisation. Normal ordered.

Tuesday 2008-11-18

Canonical transformations for fermions and bosons. Bogoliubov-Valatin transformation. Weakly interacting Bose gas. For the theory of the weakly interacting Bose gas, we use Pathria, available as an e-book via Chalmers library.

Wednesday 2008-11-19

Weakly interacting Bose gas. Non-interacting ground state not eigenstate of interacting Hamiltonian. The number of particles is not necessarily a good quantum number. Creation and annihilation operators approximated as c-numbers. Hartree term. Fock term (exchange term). Anomalous term. Bug in superfluid.

Friday 2008-11-21

The difficulty in handling lots of particles. Landau-Ginzburg theory. Global phase transformation is a symmetry: Goldstone's theorem. What we mean by momentum. Wick's theorem.

Tuesday 2008-11-25

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Wednesday 2008-11-26

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Friday 2008-11-28

Not too much here, the interesting stuff are on the slides used in this lecture.

Tuesday 2008-12-02

New lecturer: Bernhard Mehlig. Bohr's model of the hydrogen atom. De Broglie waves and classical paths. Pauli's Ph.D. thesis [from the description, I'm guessing it is this one]. The Schrödinger equation for a free particle: Hamilton's principal function. The correspondence principle. The group velocity is the classical velocity. The connection between the time-independent wave-function and the Maupertuis action. The Maupertuis principle δS=0. Fermat's principle. Classical mechanics: Lagrangian mechanics. Lagrange function. Variational principle and Lagrange's equations. Hamiltonian mechanics and Hamilton's equations. The Hamilton-Jacobi equation.

Wednesday 2008-12-03

A particle bound in a potential. Writing the Schrödinger equation in terms of the classical momentum p(x) instead of energy and potential V(x). Expanding the action S(x) in powers of ħ. The WKB wave function. Borel-resummation. The harmonic oscillator in the WKB approximation. Turning points. Patching with a linearised potential. The exact solution of the linearised problem.

Friday 2008-12-05

The stationary phase approximation. Fresnel integrals. Patching with the Airy function. Matching the wave-functions in the middle. Path integrals in phase space. The WKB quantisation condition. Maslov index. Turning points. Caustics.

Tuesday 2008-12-09

WKB quantisation. Quantizing the Harmonic oscillator. The Maslov index. Area in phase space. In several dimensions: Action-angle coordinates. Tori in phase space. Scattering. ...

Wednesday 2008-12-10

Scattering. ...

Friday 2008-12-12

Adiabatic theorem. Berry's phase. ...