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 Dominique Mouhanna and Frédéric Van Wijland.

Dominique Mouhanna’s research stands at the boundary between field theory and condensed matter physics. His main activity is to use nonperturbative renormalization group techniques in order to elucidate the critical, and more generally, the long distance behaviour of systems coming from statistical, condensed matter, and soft matter physics.

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.