Quantum Mechanics; An Introduction

Contents
1 The Quantization of Physical Quantities
1.1 Light Quanta
1.2 The Photoelectric Effect
1.3 The Compton Effect
1.4 The Ritz Combination Principle
1.5 The Franck-Hertz Experiment
1.6 The Stern-Gerlach Experiment
2 The Radiation Laws
2.1 A Preview of the Radiation of Bodies
2.2 What is Cavity Radiation
Ex. 2.1 On Cavity Radiation
2.3 The Rayleigh-Jeans Radiation Law: The Electromagnetic Eigenmodes of a Cavity
2.4 Planck's Radiation Law
Ex. 2.2 The Derivation of Planck's Radiation Law According to Planck
Ex. 2.3 Black Body Radiation
Ex. 2.4 Wien's Displacement Law
Ex. 2.5 Emmitted Energies of a Black Body
Ex. 2.6 Cosmic Black Body Radiation
3 Wave Aspects of Matter
3.1 De Broglie Waves
3.2 The Diffraction of Matter Waves
Ex. 3.1 Diffraction Patterns Generated by Monochromatic X-rays
Ex. 3.2 Scattering of Electrons and Neutrons
3.3 The Statistical Interpretation of Matter Waves
3.4 Mean (Expectation) Values in Quantum Mechanics
3.5 Three Quantum-Mechanical Operators
Ex. 3.3 The Expectation Value of Kinetic Energy
3.6 The Superposition Principle in Quantum Mechanics
Ex. 3.4 Superposition of Plane Waves, Momentum Probability
3.7 The Heisenberg Uncertainty Principle
Ex. 3.5 Position Measurement with a Slit
Ex. 3.6 Position Measurement by Enclosing a Particle in a Box
Ex. 3.7 Position Measurement with a Microscope
Ex. 3.8 Momentum Measurement with a Diffraction Grating
Ex. 3.9 Physical Supplement: The Resolving Power of a Grating
Ex. 3.10 Properties of a Gaussian Wave Packet
Ex. 3.12 Melons in Quantum Land
4 Mathematical Foundations of Quantum Mechanics I
4.1 Properties of Operators
4.2 Combining Two Operators
4.3 Bra and Ket Notation
4.4 Eigenvalues and Eigenfunctions
Ex. 4.1 Hermiticity of the Momentum Operator
Ex. 4.2 The Commutator of Position and Momentum Operators
Ex. 4.3 Computation Rules for Commutators
Ex. 4.4 Momentum Eigenfunctions
4.5 Measurability of Different Observables at Equal Times
4.6 Position and Momentum Operators
4.7 Heisenberg's Uncertainty Relations for Arbitrary Observables
4.8 Angular-Momentum Operators
4.9 Kinetic Energy
4.10 Total Energy
Ex. 4.5 Proof of an Operator Inequality
Ex. 4.6 The difference Between Uncertainty Relations
Ex. 4.7 Expansion of an Operator
Ex. 4.8 Legendre Polynomials
Ex. 4.9 Mathematical Supplement: Spherical Harmonics
Ex. 4.10 The Addition Theorem of Spherical Harmonics
5 Mathematical Supplement
5.1 Eigendifferentials and the Normalization of Eigenfunctions for Continuous Spectra
5.2 Expansion into Eigenfunctions
Ex. 5.1 Normalization of the Eigenfunctions of the Momentum Operator p_(x)
Ex. 5.2 A representation of the \delta function
Ex. 5.3 Cauchy's Principal Value
Ex. 5.4 The \delta Function as the Limit of Bell-Shaped Curves
6 The Schrödinger Equation
Ex. 6.1 A Particle in an Infinitely High Potential Well
Ex. 6.2 A Particle in a One-Dimensional Finite Potential Well
Ex. 6.3 The Delta Potential
Ex. 6.4 Distribution Functions in Quantum Statistics
Ex. 6.5 The Fermi Gas
Ex. 6.6 An Ideal Classical Gas
Ex. 6.7 A Particle in a Two-Centred Potential
6.1 The Conservation of Particle Number in Quantum Mechanics
6.2 Stationary States
6.3 Properties of Stationary States
Ex. 6.8 Current Density of a Spherical Wave
Ex. 6.9 A Particle in a Periodic Potential
7 The Harmonic Oscillator
Ex. 7.1 Mathematical Supplement: Hypergeometric Functions
7.1 The Solution of the Oscillator Equation
Ex. 7.2 Mathematical Supplement: Hermite Polynomials
7.2 The Description of the Harmonic Oscillator by Creation and Annihilation Operators
7.3 Properties of the Operators a and a+
7.4 Represantation of the Oscillator Hamiltonian in Terms of a and a+
7.5 Interpretation of a and a+
Ex. 7.3 The Three-Dimensional Harmonic Oscillator
8 The Transition from Classical to Quantum Mechanics
8.1 Motion of the Mean Values
8.2 Ehrenfest's Theorem
8.3 Constants of Motion, Laws of Conservation
Ex. 8.1 Commutation Relations
Ex. 8.2 The Virial Theorem
8.4 Quantization in Curvilinear Coordinates
Ex. 8.3 The Kinetic-Energy Operator in Spherical Coordinates
Ex. 8.4 Review of Some Useful Relations of Classical Mechanics Lagrange and Poisson Brackets
9 Charged Particles in Magnetic Fields
9.1 Coupling to the Electromagnetic Field
Ex. 9.1 The Hamilton Equations in an Electromagnetic Field
Ex. 9.2 The Lagrangian and Hamiltonian of a Charged Particle
Ex. 9.3 Landau States
9.2 The Hydrogen Atom
9.5 Currents in the Hydrogen Atom
9.6 The Magnetic Moment
Ex. 9.4 The Angular-Dependent Part of the Hydrogen Wave Function
12 Spin
12.1 Doublet Splitting
12.2 The Einstein-de Haas Experiment
12.3 The Mathematical Description of Spin
12.4 Wave Functions with Spin
12.5 The Pauli Equation
Ex. 12.1 Spin Precession in a Homogeneuos Magnetic Field
Ex. 12.2 The Rabi Experiment (Spin Resonance)
12.3 The Simple Zeeman Effect (Weak Magnetic Fields)