Visual Quantum Mechanics

Table of Contents (Book One): Sections and Subsections

>> Show only the chapters

Cover Text
Preface
1. Visualization of Wave Functions
- 1.1 Introduction
- 1.2 Visualization of Complex Numbers
  - 1.2.1 The two-dimensional manifold of complex numbers
  - 1.2.2 The three-dimensional color manifold
  - 1.2.3 A color code for complex numbers
- 1.3 Visualization of Complex-Valued Functions
  - 1.3.1 Complex-valued functions in one dimension
  - 1.3.2 Higher dimensional wave functions
- 1.4 Special Topic: Wave Functions with an Inner Structure
2. Fourier Analysis of Wave Functions
- 2.1 Fourier Series of Complex-Valued Functions
  - 2.1.1 Basic definitions
  - 2.1.2 Fourier expansion of square-integrable functions
  - 2.1.3 The convergence of the Fourier series
- 2.2 The Hilbert Space of Square-Integrable Functions
  - 2.2.1 Linear structure
  - 2.2.2 Norm and scalar product
  - 2.2.3 Other Hilbert spaces
  - 2.2.4 Orthogonality
  - 2.2.5 Fourier series
- 2.3 The Fourier Transformation
  - 2.3.1 From the Fourier series to the Fourier integral
  - 2.3.2 Fourier transformation in $n$ dimensions
- 2.4 Basic Properties of the Fourier Transform
  - 2.4.1 Riemann-Lebesgue lemma
  - 2.4.2 Fourier-Plancherel theorem
- 2.5 Linear Operators
  - 2.5.1 Basic definitions
  - 2.5.2 Boundedness
  - 2.5.3 Special topic: Continuity
  - 2.5.4 Special topic: Extension of the Fourier transform
- 2.6 Further Results About the Fourier Transformation
  - 2.6.1 Translation, phase shift, scaling transformation
  - 2.6.2 Derivative, multiplication
  - 2.6.3 Special topic: Generalized derivative
- 2.7 Gaussian Functions
  - 2.7.1 The Fourier transform of a Gaussian function
  - 2.7.2 Special topic: A dense set of Gaussian functions
- 2.8 Inequalities
  - 2.8.1 The uncertainty relation
  - 2.8.2 Special topic: Sobolev and Hardy inequalities
- 2.9 Special Topic: Dirac Delta Distribution
3. Free Particles
- 3.1 The Free Schrödinger Equation
  - 3.1.1 Matter waves
  - 3.1.2 The Schrödinger equation
  - 3.1.3 Scaling the unit of length
  - 3.1.4 Plane waves
- 3.2 Wave Packets
  - 3.2.1 Superpositions of plane waves
  - 3.2.2 Continuous superposition
  - 3.2.3 Fourier transformation
  - 3.2.4 Example: Gaussian amplitude function
- 3.3 The Free Time Evolution
  - 3.3.1 Solution of the Schrödinger equation
  - 3.3.2 Example: Gaussian function
  - 3.3.3 Conservation of the norm
  - 3.3.4 The propagator
- 3.4 The Physical Meaning of a Wave Function
  - 3.4.1 Interpretation of the wave function
  - 3.4.2 Elementary measurements
  - 3.4.3 Expectation value
  - 3.4.4 The measurement process
- 3.5 Continuity Equation
- 3.6 Special Topic: Asymptotic Time Evolution
- 3.7 Schrödinger Cat States
  - 3.7.1 Superposition of two Gaussian functions
  - 3.7.2 Schrödinger's cat
- 3.8 Special Topic: Energy Representation
4. States and Observables
- 4.1 The Hilbert Space of Wave Functions
  - 4.1.1 State vectors
  - 4.1.2 Superposition principle
- 4.2 Observables and Linear Operators
  - 4.2.1 The position operator
  - 4.2.2 Momentum operator
  - 4.2.3 Kinetic energy
- 4.3 Expectation value of an observable
- 4.4 Other observables
  - 4.4.1 The substitution rule
  - 4.4.2 Functions of x
  - 4.4.3 Functions of p
  - 4.4.4 Angular momentum
- 4.5 The commutator of x and p
- 4.6 Electromagnetic Fields
  - 4.6.1 Electric Potentials
  - 4.6.2 Magnetic Fields
- 4.7 Gauge Fields
  - 4.7.1 Non-uniqueness of the wave function
  - 4.7.2 Non-uniqueness of the Schrödinger equation
  - 4.7.3 Gauge transformations of magnetic fields:
  - 4.7.4 Gauge transformation of the Schrödinger equation
- 4.8 Projection Operators
  - 4.8.1 An example
  - 4.8.2 Measurements
  - 4.8.3 The general projection postulate
- 4.9 Transition probability
5. Boundary Conditions
- 5.1 Impenetrable Barrier
  - 5.1.1 Dirichlet Boundary Conditions
  - 5.1.2 Plane waves
  - 5.1.3 Wave packets
  - 5.1.4 Reflection of a Gaussian wave packet
- 5.2 Other Boundary Conditions
- 5.3 Particle in a Box
  - 5.3.1 Gaussian wave packet between two walls
  - 5.3.2 Method of mirrors
  - 5.3.3 A special set of solutions
- 5.4 Eigenvalues and Eigenfunctions
  - 5.4.1 Eigenvectors of linear operators
  - 5.4.2 Eigenfunctions in a box
  - 5.4.3 Time dependence of eigenfunctions
  - 5.4.4 Eigenfunction expansion
- 5.5 An Example: Unit Function in a Dirichlet Box
- 5.6 Particle on a Circle
- 5.7 The Double Slit Experiment
  - 5.7.1 The Experimental Setup
  - 5.7.2 Quantum Mechanical Description
  - 5.7.3 Comparing Theory with Experiment
  - 5.7.4 The Predictions of Quantum Mechanics
- 5.8 Special Topic: Analysis of the Double Slit Experiment
  - 5.8.1 Events and Probability
  - 5.8.2 Classical Consideration
6. Linear Operators in Hilbert Spaces
- 6.1 Hamiltonian and Time Evolution
- 6.2 Unitary Operators
- 6.3 Unitary Time Evolution, Unitary Groups
- 6.4 Symmetric Operators
- 6.5 The Adjoint Operator
  - 6.5.1 Adjoint of a bounded operator
  - 6.5.2 Adjoint of a unitary operator
  - 6.5.3 Special Topic: Adjoint of an unbounded operator
- 6.6 Self-Adjointness, Stone's Theorem
- 6.7 Translation Group
  - 6.7.1 Translations
  - 6.7.2 Translations in momentum space
- 6.8 Weyl Relations
- 6.9 Canonical Commutation Relations
- 6.10 Commutator and Uncertainty Relation
- 6.11 Symmetries and Conservation Laws
7. Harmonic Oscillator
- 7.1 Basic Definitions and Properties
  - 7.1.1 Classical Mechanics
  - 7.1.2 Quantum Mechanics
  - 7.1.3 Scaling transformation of the Hamiltonian
  - 7.1.4 Dimensionless units
  - 7.1.5 Orders of magnitude
- 7.2 Eigenfunction Expansion
  - 7.2.1 Eigenvalues of the Hamiltonian
  - 7.2.2 Expansion into eigenfunctions
  - 7.2.3 Comparison with Classical Motion
- 7.3 Solution of the Initial Value Problem
  - 7.3.1 The time evolution
  - 7.3.2 Periodic time dependence
  - 7.3.3 Fourier Transform of Oscillator States
- 7.4 Time Evolution of Observables
  - 7.4.1 Time-dependence of operators
  - 7.4.2 Position and Momentum Observables
  - 7.4.3 Time Evolution and Translation
- 7.5 Motion of Gaussian Wave Packets
  - 7.5.1 Coherent states
  - 7.5.2 Arbitrary Gaussian function
- 7.6 Harmonic Oscillator in Two and More Dimensions
- 7.7 Theory of the Harmonic Oscillator
  - 7.7.1 Supersymmetry
  - 7.7.2 The Spectrum of Eigenvalues
  - 7.7.3 The Eigenvectors
- 7.8 More about Coherent States
  - 7.8.1 Coherent states
  - 7.8.2 Completeness of Oscillator Eigenfunctions
- 7.9 Special Topic: Mehler Kernel
8. Special Systems
- 8.1 The Free Fall in a Constant Force Field
  - 8.1.1 Classical mechanics
  - 8.1.2 The quantum time evolution
  - 8.1.3 Position and momentum operators
- 8.2 Free Fall with Elastic Reflection at the Ground
- 8.3 Magnetic Fields in Two Dimensions
- 8.4 Constant Magnetic Field
  - 8.4.1 The Schrödinger equation
  - 8.4.2 The velocity operators
- 8.5 Energy Spectrum in a Constant Magnetic Field
- 8.6 Translational Symmetry in a Magnetic Field
  - 8.6.1 Classical motion
  - 8.6.2 Symmetry under translations
  - 8.6.3 Infinite degeneracy of eigenvalues
  - 8.6.4 Translation preserving the center of motion
- 8.7 Time Evolution in a Constant Magnetic Field
  - 8.7.1 Time-dependence of the quantum mechanical operators
  - 8.7.2 Motion on circles
  - 8.7.3 Rotational symmetry
  - 8.7.4 Unitary time evolution
- 8.8 Systems with Rotational Symmetry in Two Dimensions
  - 8.8.1 Rotations
  - 8.8.2 Polar coordinates
  - 8.8.3 Eigenvalue problem in polar coordinates
- 8.9 Spherical Harmonic Oscillator
- 8.10 Angular Momentum Eigenstates in a Magnetic Field
9. One-Dimensional Scattering Theory
- 9.1 Asymptotic Behavior
- 9.2 Example: Potential Step
  - 9.2.1 Continuity condition
  - 9.2.2 Energies higher than the step size
  - 9.2.3 Total reflection
  - 9.2.4 Scattering from the right
- 9.3 Wave Packets, Eigenfunction Expansion
  - 9.3.1 Energy representation in a constant potential
  - 9.3.2 Wave packets in a step potential
- 9.4 Potential Step: Asymptotic Momentum Distribution
- 9.5 Scattering Matrix
- 9.6 Transition Matrix, Several Steps
- 9.7 The Tunnel Effect
- 9.8 Example: Potential Well
  - 9.8.1 Bound state energies
  - 9.8.2 Energy spectrum for the potential well
  - 9.8.3 The scattering matrix
- 9.9 Parity
  - 9.9.1 The parity transformation
  - 9.9.2 Example: The rectangular well
Appendix A. Numerical Solution in One Dimension
- 1.0.3 Discretization of the Schrödinger equation
- 1.0.4 Solution of a linear equation with tridiagonal matrix
- 1.0.5 Crank-Nicolson method for the Schrödinger equation
- 1.0.6 Discussion
Appendix B. Movie Index
- 1 Visualization
- 2 Fourier Analysis
- 3 Free Particles
- 4 Boundary Conditions
- 5 Harmonic Oscillator
- 6 Special Systems
- 7 Scattering Theory
Appendix C. Other Books on Quantum Mechanics
Bibliography
Index
Color Plates




Table of Contents (Book One): Sections and Subsections >> Show only the chapters Cover Text Preface 1. Visualization of Wave Functions 1.1 Introduction 1.2 Visualization of Complex Numbers 1.2.1 The two-dimensional manifold of complex numbers 1.2.2 The three-dimensional color manifold 1.2.3 A color code for complex numbers 1.3 Visualization of Complex-Valued Functions 1.3.1 Complex-valued functions in one dimension 1.3.2 Higher dimensional wave functions 1.4 Special Topic: Wave Functions with an Inner Structure 2. Fourier Analysis of Wave Functions 2.1 Fourier Series of Complex-Valued Functions 2.1.1 Basic definitions 2.1.2 Fourier expansion of square-integrable functions 2.1.3 The convergence of the Fourier series 2.2 The Hilbert Space of Square-Integrable Functions 2.2.1 Linear structure 2.2.2 Norm and scalar product 2.2.3 Other Hilbert spaces 2.2.4 Orthogonality 2.2.5 Fourier series 2.3 The Fourier Transformation 2.3.1 From the Fourier series to the Fourier integral 2.3.2 Fourier transformation in $n$ dimensions 2.4 Basic Properties of the Fourier Transform 2.4.1 Riemann-Lebesgue lemma 2.4.2 Fourier-Plancherel theorem 2.5 Linear Operators 2.5.1 Basic definitions 2.5.2 Boundedness 2.5.3 Special topic: Continuity 2.5.4 Special topic: Extension of the Fourier transform 2.6 Further Results About the Fourier Transformation 2.6.1 Translation, phase shift, scaling transformation 2.6.2 Derivative, multiplication 2.6.3 Special topic: Generalized derivative 2.7 Gaussian Functions 2.7.1 The Fourier transform of a Gaussian function 2.7.2 Special topic: A dense set of Gaussian functions 2.8 Inequalities 2.8.1 The uncertainty relation 2.8.2 Special topic: Sobolev and Hardy inequalities 2.9 Special Topic: Dirac Delta Distribution 3. Free Particles 3.1 The Free Schrödinger Equation 3.1.1 Matter waves 3.1.2 The Schrödinger equation 3.1.3 Scaling the unit of length 3.1.4 Plane waves 3.2 Wave Packets 3.2.1 Superpositions of plane waves 3.2.2 Continuous superposition 3.2.3 Fourier transformation 3.2.4 Example: Gaussian amplitude function 3.3 The Free Time Evolution 3.3.1 Solution of the Schrödinger equation 3.3.2 Example: Gaussian function 3.3.3 Conservation of the norm 3.3.4 The propagator 3.4 The Physical Meaning of a Wave Function 3.4.1 Interpretation of the wave function 3.4.2 Elementary measurements 3.4.3 Expectation value 3.4.4 The measurement process 3.5 Continuity Equation 3.6 Special Topic: Asymptotic Time Evolution 3.7 Schrödinger Cat States 3.7.1 Superposition of two Gaussian functions 3.7.2 Schrödinger's cat 3.8 Special Topic: Energy Representation 4. States and Observables 4.1 The Hilbert Space of Wave Functions 4.1.1 State vectors 4.1.2 Superposition principle 4.2 Observables and Linear Operators 4.2.1 The position operator 4.2.2 Momentum operator 4.2.3 Kinetic energy 4.3 Expectation value of an observable 4.4 Other observables 4.4.1 The substitution rule 4.4.2 Functions of x 4.4.3 Functions of p 4.4.4 Angular momentum 4.5 The commutator of x and p 4.6 Electromagnetic Fields 4.6.1 Electric Potentials 4.6.2 Magnetic Fields 4.7 Gauge Fields 4.7.1 Non-uniqueness of the wave function 4.7.2 Non-uniqueness of the Schrödinger equation 4.7.3 Gauge transformations of magnetic fields: 4.7.4 Gauge transformation of the Schrödinger equation 4.8 Projection Operators 4.8.1 An example 4.8.2 Measurements 4.8.3 The general projection postulate 4.9 Transition probability 5. Boundary Conditions 5.1 Impenetrable Barrier 5.1.1 Dirichlet Boundary Conditions 5.1.2 Plane waves 5.1.3 Wave packets 5.1.4 Reflection of a Gaussian wave packet 5.2 Other Boundary Conditions 5.3 Particle in a Box 5.3.1 Gaussian wave packet between two walls 5.3.2 Method of mirrors 5.3.3 A special set of solutions 5.4 Eigenvalues and Eigenfunctions 5.4.1 Eigenvectors of linear operators 5.4.2 Eigenfunctions in a box 5.4.3 Time dependence of eigenfunctions 5.4.4 Eigenfunction expansion 5.5 An Example: Unit Function in a Dirichlet Box 5.6 Particle on a Circle 5.7 The Double Slit Experiment 5.7.1 The Experimental Setup 5.7.2 Quantum Mechanical Description 5.7.3 Comparing Theory with Experiment 5.7.4 The Predictions of Quantum Mechanics 5.8 Special Topic: Analysis of the Double Slit Experiment 5.8.1 Events and Probability 5.8.2 Classical Consideration 6. Linear Operators in Hilbert Spaces 6.1 Hamiltonian and Time Evolution 6.2 Unitary Operators 6.3 Unitary Time Evolution, Unitary Groups 6.4 Symmetric Operators 6.5 The Adjoint Operator 6.5.1 Adjoint of a bounded operator 6.5.2 Adjoint of a unitary operator 6.5.3 Special Topic: Adjoint of an unbounded operator 6.6 Self-Adjointness, Stone's Theorem 6.7 Translation Group 6.7.1 Translations 6.7.2 Translations in momentum space 6.8 Weyl Relations 6.9 Canonical Commutation Relations 6.10 Commutator and Uncertainty Relation 6.11 Symmetries and Conservation Laws 7. Harmonic Oscillator 7.1 Basic Definitions and Properties 7.1.1 Classical Mechanics 7.1.2 Quantum Mechanics 7.1.3 Scaling transformation of the Hamiltonian 7.1.4 Dimensionless units 7.1.5 Orders of magnitude 7.2 Eigenfunction Expansion 7.2.1 Eigenvalues of the Hamiltonian 7.2.2 Expansion into eigenfunctions 7.2.3 Comparison with Classical Motion 7.3 Solution of the Initial Value Problem 7.3.1 The time evolution 7.3.2 Periodic time dependence 7.3.3 Fourier Transform of Oscillator States 7.4 Time Evolution of Observables 7.4.1 Time-dependence of operators 7.4.2 Position and Momentum Observables 7.4.3 Time Evolution and Translation 7.5 Motion of Gaussian Wave Packets 7.5.1 Coherent states 7.5.2 Arbitrary Gaussian function 7.6 Harmonic Oscillator in Two and More Dimensions 7.7 Theory of the Harmonic Oscillator 7.7.1 Supersymmetry 7.7.2 The Spectrum of Eigenvalues 7.7.3 The Eigenvectors 7.8 More about Coherent States 7.8.1 Coherent states 7.8.2 Completeness of Oscillator Eigenfunctions 7.9 Special Topic: Mehler Kernel 8. Special Systems 8.1 The Free Fall in a Constant Force Field 8.1.1 Classical mechanics 8.1.2 The quantum time evolution 8.1.3 Position and momentum operators 8.2 Free Fall with Elastic Reflection at the Ground 8.3 Magnetic Fields in Two Dimensions 8.4 Constant Magnetic Field 8.4.1 The Schrödinger equation 8.4.2 The velocity operators 8.5 Energy Spectrum in a Constant Magnetic Field 8.6 Translational Symmetry in a Magnetic Field 8.6.1 Classical motion 8.6.2 Symmetry under translations 8.6.3 Infinite degeneracy of eigenvalues 8.6.4 Translation preserving the center of motion 8.7 Time Evolution in a Constant Magnetic Field 8.7.1 Time-dependence of the quantum mechanical operators 8.7.2 Motion on circles 8.7.3 Rotational symmetry 8.7.4 Unitary time evolution 8.8 Systems with Rotational Symmetry in Two Dimensions 8.8.1 Rotations 8.8.2 Polar coordinates 8.8.3 Eigenvalue problem in polar coordinates 8.9 Spherical Harmonic Oscillator 8.10 Angular Momentum Eigenstates in a Magnetic Field 9. One-Dimensional Scattering Theory 9.1 Asymptotic Behavior 9.2 Example: Potential Step 9.2.1 Continuity condition 9.2.2 Energies higher than the step size 9.2.3 Total reflection 9.2.4 Scattering from the right 9.3 Wave Packets, Eigenfunction Expansion 9.3.1 Energy representation in a constant potential 9.3.2 Wave packets in a step potential 9.4 Potential Step: Asymptotic Momentum Distribution 9.5 Scattering Matrix 9.6 Transition Matrix, Several Steps 9.7 The Tunnel Effect 9.8 Example: Potential Well 9.8.1 Bound state energies 9.8.2 Energy spectrum for the potential well 9.8.3 The scattering matrix 9.9 Parity 9.9.1 The parity transformation 9.9.2 Example: The rectangular well Appendix A. Numerical Solution in One Dimension 1.0.3 Discretization of the Schrödinger equation 1.0.4 Solution of a linear equation with tridiagonal matrix 1.0.5 Crank-Nicolson method for the Schrödinger equation 1.0.6 Discussion Appendix B. Movie Index 1 Visualization 2 Fourier Analysis 3 Free Particles 4 Boundary Conditions 5 Harmonic Oscillator 6 Special Systems 7 Scattering Theory Appendix C. Other Books on Quantum Mechanics Bibliography Index Color Plates