Contents

• Introduction
  • The Radiation Laws and the Birth of Quantum Mechanics
    • Remarks
    • * Math: Density of states
  • Waves, particles, and wave packets
    • Introduction
    • The wave packet
  • Interpretation of the Wave Function
    • Interference experiments
    • First Axiom
    • Probability densities
    • Example: Gauss distribution
    • Example: distribution
    • Expectation value (mean value), mean square deviation
    • The continuity equation
  • Fourier Transforms and the Solution of the Schrödinger Equation
    • Math: Fourier Integral
    • Math: Gauss function
    • The Delta Functional (`Delta Function')
    • * Partial Differential Equations and Fourier Transform
  • Position and Momentum in Quantum Mechanics
    • Expectation values in quantum mechanics
    • Example: Wave packet
    • The commutator
  
  
• Wave Mechanics
  • The Stationary Schrödinger Equation
    • Stationary states
    • Math Revision: Eigenvalues of a Two-by-Two-Matrix
    • Stationary States in One Dimension
    • Piecewise Constant one-Dimensional Potentials
  • The Infinite Potential Well
    • Wave functions and eigenenergies
    • The Hilbert space of wave functions
  • The Potential Well
    • Wave functions and energy eigenvalues for
    • The parity
    • Even and odd solutions for the potential well
  • Scattering states in one dimension
    • Plane waves
    • Potential scattering
  • The Tunnel Effect and Scattering Resonances
    • Potential step
    • The Tunnel Barrier: Transmission Coefficient
    • The tunnel barrier: Discussion
  • A more complicated case
    • Case of no potential
    • Tunnel barrier potential within well
  
  
• The Structure of Quantum Mechanics
  • Axioms of Quantum Mechanics and the Hilbert Space
    • Recalling our axioms
    • Math: The Hilbert Space
    • Math: Examples of Hilbert spaces
    • First Axiom: States as Hilbert Space Vectors
    • Math: Completeness, Dirac notation
  • Operators and The Two-Level-System I
    • Operators
    • Examples for Expectation Values
    • Matrix Operators and The Two-Level System
  • Operators and Measurements
    • Math: Linear Operators
    • Eigenvalues and Measurement
    • Example: Position Measurement in the Two-Level system
  • Energy Measurements
    • Eigenstates of the Two-Level System
    • Energy Measurement in the Two-Level System
    • *Experiments in Double Quantum Dots
    • Energy Measurement in a Quantum Well
  • The Two-Level System: Time-Evolution
    • Time Evolution of States
  
  
• Important Quantum Mechanical Model Systems
  • The Harmonic Oscillator I
    • Model
    • Solution of the Differential Equation
  • The Harmonic Oscillator II
    • The Hilbert space of wave functions
    • Discussion: Energy Eigenvalues
    • Harmonic Oscillator: Expectation Values
  • Ladder Operators, Phonons and Photons
    • The Ladder Operators and
    • The Harmonic Oscillator
    • Phonons and Photons
  • The Hydrogen Atom
    • Spherical Symmetric Potentials in 3 Dimensions
    • The Angular Part
    • The Radial Part
  • The Angular Momentum
    • Definitions
  
  
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