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.