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 2: A Particle in a Spherically Symmetric Potential

$1: The two-body problem
1.1 Two interacting particles
1.2 Reduction to a one-body problem
1.3 Separation of the Schrödinger equation for the relative motion


$2: Angular dependence, spherical harmonics
2.1 Separation of the operator Lambda
2.2 Legendre polynomials
2.3 Associated Legendre functions
2.4 Spherical harmonics

$3: A free particle with a particular angular momentum
3.1 The radial equation
3.2 The radial equation for a free particle
3.3 A free particle with l = 0
3.4 Solution for l > 0
3.5 Asymptotic behavior of the wave function

$4: The spherical potential well
4.1 Bound states in a deep well
4.2 s-states in a well of finite depth


$5: The 3-d harmonic oscillator
5.1 The radial equation
5.2 Laguerre polynomials
5.3 The energy spectrum


$6: The Coulomb problem
6.1 The radial equation
6.2 The energy spectrum of the bound states
6.3 The wave functions of the bound states

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

$1: Algebraic preliminaries
1.1 Metric spaces
1.2 Banach spaces
1.3 Hilbert spaces


$2: The axioms of QM, and elements of measurement theory
2.1 Hilbert space and QM
2.2 Projectors and probabilities
2.3 Multiple measurements


$3: Elements of representation theory
3.1 Representation of states
3.2 Change of representation
3.3 Representation of operators

$4: Unitary transformations
4.1 Unitary operators
4.2 Time evolution as a unitary transformation
4.3 The Schrödinger, Heisenberg, and Interaction Pictures of QM

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
1.5 Addition of angular momenta

$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