- List of topics of QM-II, QM-III and QM-Advanced courses.
- General checklist of topics in Quantum Mechanics.

- Dirac notations, the continuum limit.

- The fundemental postulates of Quantum mechanics

- Magentic field (gauge, AB, Landau, Hall)

- Group theory, Representations, Rotations, Spin
- Addition of angular momentum (elementary level)

QM-II topics / second part

- Perturbation theory for eigenstates (up to 2nd order)
- Perturbation theory for time evolution (first order)

- Wigner decay, FGR transtions, scattering by Born formula

- The adiabatic picture (elementary level)

**QM-III topics / first part ****(**For more details see below**)**

- Path integrals

- The resolvent and Green functions

- Perturbation theory to infinite order

- Semiclassical approximations (brief).
- Scattering theory (S matrix, partial waves).

QM-III topics / second part

- Transformations of H and invariance
- Addition of angular momentum etc.
- Identical particles (elementary level + Fock space)
- Quantization of the EM field

**
QM - advanced topics **

- The probability matrix

- Theory of quantum measurements
- Wigner function and Wigner-Weyl formalism

- Schrodinger cat, EPR, Bell...

- Quantum Computation.
- Adiabatic processes. Berry phase.
- Linear response theory (Kubo formula).
- Born-Oppenheimer picture.

**Elaborated list of topic for the first part of the QM-III course:**

- The evolution operator for constant H
- diagonalization
- space representation U(x|x0)
- for free particle
- for bounded particle (oscillator)
- for short times
- Feynman path integral
- Semiclassical evaluation

- The resolvent G(z)
- for closed systems
- for open system (analytical continuation)
- Green functions
- for free particle in 1D/2D/3D
- Green theorem

- perturbation theory for G(z)
- perturbation theory for U

- perturbation theory for scattering states
- the asymptotic wave function
- cross section
- S matrix via the T expansion

- the S-matrix formalism
- S matrix for a wire with delta function
- S matrix for several connected wires / leads

- S matrix for spherical geometry
- unitarity and optical theorem

- phase shifts
- scattering from a sphere / well

- resonances

- Born approximation for the phase shifts

- Fisher Lee relation
- the R matrix approach

- Banded matrices (Wigner/Anderson models)
- Random matrix theory and Quantum Chaos

- Theory of conductance and quantum pumping

- Born-Oppenheimer approximation
- Feynman-Vernon Formalism
- Master equations
- Dissipation and Decoherence

- Two level system - The spin boson model
- Multi level system - Pauli Eq.
- Damped particle - Fokker-Plank Eq.
- Reaction rate theory, Kramer

Below is the "maximal" list of topics with which one should be familiar at the end of the QM-II course.

`(*) advanced topics, if there is time`
`(**) excluded topics, not in the QM-II course
`

- Hilbert space
- The probability matrix (*)
- Unitary evolution

- Unitary operations:
- Displacements, Rotations, Boosts; Gauge
- The concept of invariance
- Galilei and gauge invariance (*)
- Time reversal invariance (*)
- Symmetries and their consequences (*)

- Rotations and Spin:
- Unitary representations of rotations and the spin concept.
- Actual procedure for building rotation matrices.
- Addition of angular momentum (*)

- The classical limit
- Probability currents
- Wigner function (*)

FORMALISM:

- Perturbation theory for:

- stationary states

- resolvent [Green function] (*)

- evolution operator [interaction picture]

- Dynamics [special topics]:

- wavepacket dynamics, Wigner decay

- Fermi golden rule, resonances

- Adiabatic processes:

Berry phase effect (*)

Landau-Zener transitions (*)

Born-Oppenheimer approximation (*)

- Scattering theory:

- for generalized 1D geometry

- for multi lead geometry

- for spherical geometry

[born approximation]

[cross section, optical theorem(*)]

[phase shifts, Wigner time delay(*)]

- Semiclassical methods: (*)

- Weyl formula

- WKB wavefunctions

- ergodic wavefunctions

APPLICATIONS:

- Particle in a few site system (in particular "double well").
- Particle in a rectangular box with various boundary conditions.
- Particle in a spherically symmetric potential.

- Special potentials (**elementary): Harmonic; Barrier [tunneling]; Delta; Step
- Special potentials (**advanced ): Periodic [Bloch]; Disorder [localization].

- Particle in 1D ring + magnetic flux [AB effect, persistent currents, magnetic response]
- Particle in uniform magnetic field [Landau levels, Hall effect]

- Spin in magnetic field [precession, Zeeman effect, Stern-Gerlach]
- Spin in electric field [spin-orbit interaction]

- Theory of atoms, molecules and nuclei (**)
- Dirac equation [electrons and positrons] (**)
- Identical particles [fermions, bosons] (**)
- Quantized electromagnetic field [photons] (**)