ADVANCED QUANTUM MECHANICS

I. Introduction
II. Recap and general formulation
1. Basic and operational principles: system, state; wave function Ψ(x); space of states; superposition principle; probabilistic interpretation; temporal evolution; Schrödinger equation; stationary Schrödinger equation; operators; measurement; Heisenberg inequalities.
2. First approach to conceptual problems:
- The nature of the wave function and its ontological meaning.
- Measurement problem: incompatibility between the deterministic/unitary temporal evolution of the wave function before measurement and its nondeterministic/non unitary evolution during the measurement process ; what is a measurement in comparison with an interaction ?
- Question of the completeness of quantum mechanics.
3. Abstract formalism:
- Motivation for the need of a more general framework; Hilbert space (generalities); Dirac formalism, state; ket, operator, spectrum, bra, scalar product, dyadic product, projectors; unitary, adjoint and hermitian operator; exponential of a hermitian operator, basis completeness relation, change of basis; matrix representation – Heisenberg matrices, measurement, probabilistic interpretation, mean value of an operator, commutator of two operators, CSCO, Heisenberg inequalities.
- The example of X and P: representations x and p, relation between representations, Fourier transform.
- Systems of many particles; tensor product of Hilbert spaces; density operator; mixtures and pure states; entanglement; Schmidt decomposition; two and N particles systems; identical particles; bosons and fermions.
III. Symmetries and conservation laws
1. Introduction: symmetries in Mathematics and in Physics.
2. Symmetries and conservation laws in classical physics:
Invariance of classical equations of motion under a transformation, active and passive transformations; translational invariance in space and momentum conservation; abelian translation group; rotation invariance and angular momentum conservation; non-abelian rotation group; dynamics symmetry: Galilean invariance and consequence on the structure of the Lagrangian; symmetries in Lagrangian and Hamiltonian formalisms.
3. Symmetries and conservation laws in quantum mechanics:
- Importance of symmetry in quantum mechanics.
- Representation of a space-time transformation in the Hilbert space of states.
- Wigner theorem.
- Time-evolution; unitary evolution operator U; time-independent Hamiltonian; example of two-states system; mean values; conserved quantities; non-conservative systems; timedependent Hamiltonian; Schrödinger and Heisenberg pictures.
- Translations; translation operator; translational invariance; position and momentum operators; canonical commutation relations; wave function; application: crystalline symmetries and Bloch Theorem; wave function of a composite system.
- Rotations, rotation operator; commutation relations; rotational invariance; scalar and vectorial operators.
- Galilean invariance and dynamics of the quantum particle; representation of the Galilee group; generator; velocity operator; structure of the Hamiltonian; projectivity; superselection rule of the Galilean mass; evolution of the free particle; evolution of the particle interacting with a field.
- Gauge invariance: gauge invariance in quantum mechanics and in quantum field theory.
4. Discrete symmetries:
Parity; time reversal.
IV. Theory of the angular momentum
Recap of angular momentum in classical physics; diagonalisation of the angular momentum; representation of the rotation group; the SO(3) group; orbital momentum; spin S = J-L; spherical harmonics; the quantum rotator; the quantum top; addition of angular momenta; case of two spin 1/2; Wigner-Eckart theorem.
V. Central potential
Definition; symmetries; reduced Hamiltonian; radial Schrödinger equation; the free particle; the harmonic oscillator; the hydrogen atom; derivation via dynamic symmetry.
VI. The spin
- Abstract definition; space of states; spin 1/2 ; group theoretical interpretation: representation of the group rotation; particle in a magnetic field; the SU(2)-SO(3) homeomorphism.
- Entanglement of two spin 1/2; the EPR paradox; Bell inequalities; GHZ states; Mermin-Peres test; some words about quantum cryptography; quantum computation.
- The problem of the measurement; formulation in terms of density operator; decoherence.
VII. Identical particles
Spin-statistic theorem; bosons and fermions again; elementary bosons and fermions; matter and stability of the matter; “Pauli-Heisenberg” inequality.
VIII. Approximation methods
Variational method; stationary perturbation theory; asymptotic nature of the series; example of a two-states system; Stark effect; Zeeman effect; Fine and hyperfine structure of the hydrogen atom; Path integral formulation of quantum mechanics.
IX. Time dependent perturbation theory
Position of the problem; integral representation of the Schrödinger equation; transition amplitude; non-exponential decay-rate; quantum Zeno effect; constant perturbation; monochromatic perturbation; example of a two-states system; Rabi oscillations; case of discrete states; case of continuous states; Fermi golden rule.
X. Introduction to scattering theory
Lippmann-Schwinger equation; the integral equation for diffusion; S-matrix; partial wave expansion; Born approximation; Yukawa potential.

The advanced quantum mechanics course is given by Frédéric Van Wijland and Michael Joyce.



Frédéric van Wijland's research works lie at the crossroads of statistical mechanics and field-theoretic methods, with a view to describing the emergent collective behavior of nonequilibrium systems.