(This list is not for students, but for curriculum development)

Mathematical background see here

- Fundamentals:
- Observables and wave-function in different representations, density operator (including open systems)
- Schrodinger equation and other evolution equations, Green's function and propagator
- Spin
- Many-body wave-function, density matrices (e.g. transition)
- One-particle orbitals, Slater determinant
- Second quantization – post-HF methods and empirical Hamiltonians

- Methods:
- Variational method (including linear basis set expansion)
- Perturbation theory (including degenerate, periodic, step-like, Fermi's golden rule)
- Hartree-Fock method (including spin)
- Density functional theory
- Many-body perturbation theory

- Models:
- Free particle (multidimensional) – conceptual model
- Particle in a box (multidimensional) – conceptual model
- Particle in a uniform field – atomic motions
- Harmonic oscillator – vibrations
- Rigid rotor (at least spherically symmetric) – molecular rotations
- Hydrogen atom (nonrelativistic) – atoms
- Periodic potential (Dirac comb or Kronig-Penney model or Huckel model of trans-polyacetylene) – solids
- Two-state quantum system – conceptual model
- Landau-Zener model – nonadiabatic transitions
- Double-well potential – various transitions

- Fundamentals:
- Basics of equilibrium and nonequilibrium thermodynamics
- Hamiltonian, phase space, partition function, averaging, detailed balance
- Thermodynamic potentials
- Phase diagram, phase transitions
- Kinetic equations (at least Master equation, also Boltzmann equation)

- Models:
- Two-state quantum system – conceptual model
- Harmonic oscillator – vibrations
- Rigid rotor – molecular rotations
- Ideal gas (classical and quantum) – conceptual model
- Real gas (at least van der Waals) – gases
- Equation of state of solids (e.g. Murnaghan) – solids
- Emission and absorption of boson waves (e.g. electromagnetic waves, Einstein coefficients)
- Lattice models (e.g. lattice gas or spin model) – conceptual model

- Most common structural types of molecules and solids, crystallographic notations, bonding
- Electronic structure of atoms, state notations, periodic table
- Electronic structure of molecules (e.g. Molecular orbital theory), state notations, molecular orbital notations
- Electronic structure of solids, k-point notations
- Vibration of molecules and solids, symmetry notations
- Most common observables: ionization potential, electron affinity, bandgap, electronic and vibrational spectroscopy
- See also list of concepts for Survey of Materials