# PH 414, 415, 417: Quantum Mechanics

## AY 2011/12 (D. Belitz)

### Chapter 1: Basic Notions of Quantum Mechanics

\$1: Motivation: The insufficiency of a classical description of nature
1.1 Particles and waves
1.2 The wave nature of light
1.3 The particle nature of light
1.4 The wave nature of particles
1.5 The uncertainty principle (rough idea)
1.6 Other problems with classical physics: Atoms, and the sun

\$2: Wave mechanics ( Part 1 , Part 2 , Part 3 , Part 4 )
2.1 The wave function
2.2 The superposition principle
2.3 The statistical interpretation of the wave function
2.4 A free particle in a box
2.5 Mathematical digression: Generalized Fourier expansions
2.5.1 Orthogonal functions
2.5.2 Generalized Fourier series
2.5.3 Completeness, and Parseval's formula
2.5.4 Fourier integrals, and the delta function
2.5.5 Product sets
2.6 Average values of position and momentum
2.7 Operators
2.7.1 Motivation: The need to deal with operators
2.7.2 General properties of operators in QM
2.7.3 Operators for position, momentum, angular momentum, and kinetic energy
2.8 Digression: A crash course in complex analysis
2.8.1 Complex numbers, and functions of a complex variable
2.8.2 Analytic functions
2.8.3 Path integrals
2.8.4 Laurent series
2.8.5 The residue theorem
2.8.6 Doing integrals

\$3: The formal foundations of Quantum Mechanics
3.1 States and operators
3.2 Eigenfunctions and eigenvalues
3.3 The case of discrete spectra
3.4 The case of continuous spectra
3.5 The case of mixed spectra
3.6 Simultaneously diagonal operators
3.7 The uncertainty relation
3.8 The Schrödinger equation
3.9 Stationary states

\$4: Simple applications ( Part 1 , Part 2 )
4.1 Potential step
4.2 Reflection and transmission coefficients
4.3 Rectangular potential barrier
4.4 Rectangular potential well
4.5 Harmonic ocscillator
4.6 Charged particle in a homogeneous electric field

### Chapter 3: Elements of measurement, representation, and transformation theory

Note: Dar Dahlen has kindly typed up part of Ch. 3; thanks, Dar!! The current version can be found here . I have edited some parts that Dar had trouble reading. If you spot any problems, please let Dar or me know.

### Chapter 4: Spin, the Pauli equation, and identical particles

\$1: Spin
1.1 Angular momentum revisited
1.2 General angular momentum operators
1.3 Spin
1.4 Spin 1/2, Pauli matrices

\$2: The Pauli equation
2.1 A classical charged particle in an electromagnetic field
2.2 The Pauli equation
2.3 Weak magnetic fields; the Zeeman effect
2.4 Strong magnetic fields; Landau levels

\$3: Identical particles
3.1 Indistinguishability of identical particles
3.2 Symmetric and antisymmetric wave functions
3.3 The symmetry principle of Quantum Mechanics

### Chapter 5: Time independent perturbation theory, the variational principle, and applications

\$1: Perturbation theory
1.1 General idea
1.2 Perturbation theory for a discrete, non-degenerate spectrum
1.3 Eigenvalues and eigenfunctions to first order
1.4 Eigenvalues to second order

\$2: The variational principle
2.1 Rayleigh-Ritz variational principle for the ground state

\$3: Applications
3.1 Variational principle for the ground state: Screening in the Helium atom
3.2 Heitler-London theory of the hydrogen molecule