Non-relativistic quantum mechanics courses
In this page you will find:
- List of topics of QM-II, QM-III and QM-Advanced courses.
- General checklist of topics in Quantum Mechanics.
It is assumed that the student has taken the introductory course QM-I.
Hence the student should be familiar with Dirac notations and with the
elementary solutions of the Schrodinger equation: Simple potenials in
one dimension; Harmonic oscillator; Hydrogen atom.
QM-II topics / first part
- 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 (For optional topics see below)
- 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
- 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
- Born approximation for the phase shifts
- Fisher Lee relation
- the R matrix approach
Elaborated list of optional topic for the QM-advanced course:
- 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
Quantum Mechanics - Checklist
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 (*)
- 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
[cross section, optical theorem(*)]
[phase shifts, Wigner time delay(*)]
- Semiclassical methods: (*)
- Weyl formula
- WKB wavefunctions
- ergodic wavefunctions
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] (**)