Topics

Unit I: Overview, Mathematical Introduction

(A) Introduction to quantum mechanics:
Scope and applicability of quantum mechanics
The Schrödinger equation

(B) History of quantum mechanics

(C) Linear vector spaces:
Definitions
Inner products
Dual spaces and Dirac notation

(C) Operators:
Basic operator rules
Classes of operators: linear, hermitian, unitary, etc.
Commutators

(D) Postulates

(E) Simple problems: particle in a box (1D and 3D), free particle

Unit II: Fundamentals

(A) Harmonic oscillator
Vibrational (IR) spectroscopy, anharmonicity, group theory

(B) Heisenberg uncertainty relations

(C) Angular momentum:
Commutation rules
Spherical harmonics
Ladder operators
Rigid rotor: a model for rotational (microwave) spectroscopy

(D) The Hydrogen atom and its electronic spectrum

Unit III: Approximate Methods

(A) Variational method:
Variational theorem
Equivalence of Raleigh-Ritz procedure and diagonalization

(B) Time-independent perturbation theory

Unit IV: Advanced Fundamentals

(A) Spin and ESR spectroscopy

(B) Degenerate perturbation theory and the Stark effect in H atom

(C) Time-dependent perturbation theory and the interaction of light with matter

Unit V: Electronic Structure and Spectroscopy

(A) The molecular Hamiltonian:
Born-Oppenheimer approximation

(B) Separation into electronic, vibrational, rotational terms

(C) Introduction to Hartree-Fock theory:
Two-electron problem
Hartree products
Antisymmetry and Slater determinants
Generalization to $N$-electrons
Self-consistent-field

(D) Introduction to correlated methods

(E) Electronic structure of atoms

(F) Electronic structure of diatomics

(G) Electronic structure of polyatomics:
Walsh's rules

(H) Rovibronic spectroscopy and the Franck-Condon approximation