Quantum Mechanics Solution Manual
© Leon van Dommelen
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1. Special Relativity [Draft]
Contents
1. Special Relativity [Draft]
1.1 Overview of Relativity
1.1.1 A note on the history of the theory
1.1.2 The mass-energy relation
1.1.3 The universal speed of light
1.1.4 Disagreements about space and time
1.2 The Lorentz Transformation
1.2.1 The transformation formulae
1.2.2 Proper time and distance
1.2.3 Subluminal and superluminal effects
1.2.4 Four-vectors
1.2.5 Index notation
1.2.6 Group property
1.3 Relativistic Mechanics
1.3.1 Intro to relativistic mechanics
1.3.2 Lagrangian mechanics
2. Mathematical Prerequisites
2.1 Complex Numbers
2.1.1 Solution mathcplx-a
2.1.2 Solution mathcplx-b
2.1.3 Solution mathcplx-c
2.1.4 Solution mathcplx-d
2.1.5 Solution mathcplx-e
2.1.6 Solution mathcplx-f
2.1.7 Solution mathcplx-g
2.1.8 Solution mathcplx-h
2.2 Functions as Vectors
2.2.1 Solution funcvec-a
2.2.2 Solution funcvec-b
2.3 The Dot, oops, INNER Product
2.3.1 Solution dot-a
2.3.2 Solution dot-b
2.3.3 Solution dot-c
2.3.4 Solution dot-d
2.3.5 Solution dot-e
2.3.6 Solution dot-f
2.3.7 Solution dot-g
2.4 Operators
2.4.1 Solution mathops-a
2.4.2 Solution mathops-b
2.4.3 Solution mathops-c
2.4.4 Solution mathops-d
2.5 Eigenvalue Problems
2.5.1 Solution eigvals-a
2.5.2 Solution eigvals-b
2.5.3 Solution eigvals-c
2.6 Hermitian Operators
2.6.1 Solution herm-a
2.6.2 Solution herm-b
2.6.3 Solution herm-c
2.6.4 Solution herm-d
2.6.5 Solution herm-e
2.6.6 Solution herm-f
2.6.7 Solution herm-g
2.6.8 Solution herm-h
2.6.9 Solution herm-i
2.7 Additional Points
2.7.1 Dirac notation
2.7.2 Additional independent variables
3. Basic Ideas of Quantum Mechanics
3.1 The Revised Picture of Nature
3.2 The Heisenberg Uncertainty Principle
3.3 The Operators of Quantum Mechanics
3.4 The Orthodox Statistical Interpretation
3.4.1 Only eigenvalues
3.4.2 Statistical selection
3.5 A Particle Confined Inside a Pipe
3.5.1 The physical system
3.5.2 Mathematical notations
3.5.3 The Hamiltonian
3.5.4 The Hamiltonian eigenvalue problem
3.5.5 All solutions of the eigenvalue problem
3.5.5.1 Solution piped-a
3.5.5.2 Solution piped-b
3.5.6 Discussion of the energy values
3.5.6.1 Solution pipee-a
3.5.6.2 Solution pipee-b
3.5.6.3 Solution pipee-c
3.5.7 Discussion of the eigenfunctions
3.5.7.1 Solution pipef-a
3.5.7.2 Solution pipef-b
3.5.7.3 Solution pipef-c
3.5.8 Three-dimensional solution
3.5.8.1 Solution pipeg-a
3.5.8.2 Solution pipeg-b
3.5.8.3 Solution pipeg-c
3.5.9 Quantum confinement
4. Single-Particle Systems
4.1 The Harmonic Oscillator
4.1.1 The Hamiltonian
4.1.2 Solution using separation of variables
4.1.2.1 Solution harmb-a
4.1.2.2 Solution harmb-b
4.1.2.3 Solution harmb-c
4.1.2.4 Solution harmb-d
4.1.3 Discussion of the eigenvalues
4.1.3.1 Solution harmc-a
4.1.3.2 Solution harmc-b
4.1.4 Discussion of the eigenfunctions
4.1.4.1 Solution harmd-a
4.1.4.2 Solution harmd-b
4.1.4.3 Solution harmd-c
4.1.5 Degeneracy
4.1.5.1 Solution harme-a
4.1.6 Noneigenstates
4.2 Angular Momentum
4.2.1 Definition of angular momentum
4.2.2 Angular momentum in an arbitrary direction
4.2.2.1 Solution angub-a
4.2.2.2 Solution angub-b
4.2.2.3 Solution angub-c
4.2.3 Square angular momentum
4.2.3.1 Solution anguc-a
4.2.3.2 Solution anguc-b
4.2.3.3 Solution anguc-c
4.2.4 Angular momentum uncertainty
4.3 The Hydrogen Atom
4.3.1 The Hamiltonian
4.3.2 Solution using separation of variables
4.3.2.1 Solution hydb-a
4.3.2.2 Solution hydb-b
4.3.2.3 Solution hydb-c
4.3.3 Discussion of the eigenvalues
4.3.3.1 Solution hydc-a
4.3.3.2 Solution hydc-b
4.3.3.3 Solution hydc-c
4.3.3.4 Solution hydc-d
4.3.4 Discussion of the eigenfunctions
4.3.4.1 Solution hydd-a
4.3.4.2 Solution hydd-b
4.3.4.3 Solution hydd-c
4.4 Expectation Value and Standard Deviation
4.4.1 Statistics of a die
4.4.1.1 Solution esda-a
4.4.1.2 Solution esda-b
4.4.1.3 Solution esda-c
4.4.1.4 Solution esda-d
4.4.2 Statistics of quantum operators
4.4.2.1 Solution esdb-a
4.4.2.2 Solution esdb-b
4.4.3 Simplified expressions
4.4.3.1 Solution esdb2-a
4.4.3.2 Solution esdb2-b
4.4.4 Some examples
4.5 The Commutator
4.5.1 Commuting operators
4.5.1.1 Solution commutea-a
4.5.2 Noncommuting operators and their commutator
4.5.3 The Heisenberg uncertainty relationship
4.5.3.1 Solution commutec-a
4.5.4 Commutator reference
4.6 The Hydrogen Molecular Ion
4.6.1 The Hamiltonian
4.6.2 Energy when fully dissociated
4.6.3 Energy when closer together
4.6.4 States that share the electron
4.6.5 Comparative energies of the states
4.6.6 Variational approximation of the ground state
4.6.6.1 Solution hione-a
4.6.7 Comparison with the exact ground state
5. Multiple-Particle Systems
5.1 Wave Function for Multiple Particles
5.1.1 Solution complex-a
5.1.2 Solution complex-b
5.2 The Hydrogen Molecule
5.2.1 The Hamiltonian
5.2.1.1 Solution hmola-a
5.2.1.2 Solution hmola-b
5.2.2 Initial approximation to the lowest energy state
5.2.2.1 Solution hmolb-a
5.2.2.2 Solution hmolb-b
5.2.3 The probability density
5.2.3.1 Solution hmolc-a
5.2.4 States that share the electrons
5.2.4.1 Solution hmold-a
5.2.4.2 Solution hmold-b
5.2.5 Variational approximation of the ground state
5.2.6 Comparison with the exact ground state
5.3 Two-State Systems
5.3.1 Solution 2state-a
5.3.2 Solution 2state-b
5.4 Spin
5.4.1 Solution spin-a
5.4.2 Solution spin-b
5.5 Multiple-Particle Systems Including Spin
5.5.1 Wave function for a single particle with spin
5.5.1.1 Solution complexsa-a
5.5.2 Inner products including spin
5.5.2.1 Solution complexsai-a
5.5.2.2 Solution complexsai-b
5.5.3 Commutators including spin
5.5.3.1 Solution complexsac-a
5.5.4 Wave function for multiple particles with spin
5.5.4.1 Solution complexsb-a
5.5.4.2 Solution complexsb-b
5.5.5 Example: the hydrogen molecule
5.5.5.1 Solution complexsc-a
5.5.6 Triplet and singlet states
5.5.6.1 Solution complexse-a
5.6 Identical Particles
5.6.1 Solution ident-a
5.6.2 Solution ident-b
5.7 Ways to Symmetrize the Wave Function
5.7.1 Solution symways-a
5.7.2 Solution symways-b
5.8 Matrix Formulation
5.8.1 Solution matfor-a
5.8.2 Solution matfor-b
5.9 Heavier Atoms
5.9.1 The Hamiltonian eigenvalue problem
5.9.2 Approximate solution using separation of variables
5.9.3 Hydrogen and helium
5.9.4 Lithium to neon
5.9.5 Sodium to argon
5.9.6 Potassium to krypton
5.9.7 Full periodic table
5.10 Pauli Repulsion
5.11 Chemical Bonds
5.11.1 Covalent sigma bonds
5.11.2 Covalent pi bonds
5.11.3 Polar covalent bonds and hydrogen bonds
5.11.4 Promotion and hybridization
5.11.5 Ionic bonds
5.11.6 Limitations of valence bond theory
6. Macroscopic Systems
6.1 Intro to Particles in a Box
6.2 The Single-Particle States
6.3 Density of States
6.4 Ground State of a System of Bosons
6.5 About Temperature
6.6 Bose-Einstein Condensation
6.6.1 Rough explanation of the condensation
6.7 Bose-Einstein Distribution
6.8 Blackbody Radiation
6.9 Ground State of a System of Electrons
6.10 Fermi Energy of the Free-Electron Gas
6.11 Degeneracy Pressure
6.12 Confinement and the DOS
6.13 Fermi-Dirac Distribution
6.14 Maxwell-Boltzmann Distribution
6.15 Thermionic Emission
6.16 Chemical Potential and Diffusion
6.17 Intro to the Periodic Box
6.18 Periodic Single-Particle States
6.19 DOS for a Periodic Box
6.20 Intro to Electrical Conduction
6.21 Intro to Band Structure
6.21.1 Metals and insulators
6.21.2 Typical metals and insulators
6.21.3 Semiconductors
6.21.4 Semimetals
6.21.5 Electronic heat conduction
6.21.6 Ionic conductivity
6.22 Electrons in Crystals
6.22.1 Bloch waves
6.22.2 Example spectra
6.22.3 Effective mass
6.22.4 Crystal momentum
6.22.5 Three-dimensional crystals
6.23 Semiconductors
6.24 The P-N Junction
6.25 The Transistor
6.26 Zener and Avalanche Diodes
6.27 Optical Applications
6.27.1 Atomic spectra
6.27.2 Spectra of solids
6.27.3 Band gap effects
6.27.4 Effects of crystal imperfections
6.27.5 Photoconductivity
6.27.6 Photovoltaic cells
6.27.7 Light-emitting diodes
6.28 Thermoelectric Applications
6.28.1 Peltier effect
6.28.2 Seebeck effect
6.28.3 Thomson effect
7. Time Evolution
7.1 The Schrödinger Equation
7.1.1 The equation
7.1.2 Solution of the equation
7.1.2.1 Solution schrodsol-a
7.1.2.2 Solution schrodsol-b
7.1.2.3 Solution schrodsol-c
7.1.3 Energy conservation
7.1.4 Stationary states
7.1.5 The adiabatic approximation
7.2 Time Variation of Expectation Values
7.2.1 Newtonian motion
7.2.2 Energy-time uncertainty relation
7.3 Conservation Laws and Symmetries
7.4 Conservation Laws in Emission
7.4.1 Conservation of energy
7.4.2 Combining angular momenta and parities
7.4.3 Transition types and their photons
7.4.4 Selection rules
7.5 Symmetric Two-State Systems
7.5.1 A graphical example
7.5.2 Particle exchange and forces
7.5.3 Spontaneous emission
7.6 Asymmetric Two-State Systems
7.6.1 Spontaneous emission revisited
7.7 Absorption and Stimulated Emission
7.7.1 The Hamiltonian
7.7.2 The two-state model
7.8 General Interaction with Radiation
7.9 Position and Linear Momentum
7.9.1 The position eigenfunction
7.9.2 The linear momentum eigenfunction
7.10 Wave Packets
7.10.1 Solution of the Schrödinger equation.
7.10.2 Component wave solutions
7.10.3 Wave packets
7.10.4 Group velocity
7.10.5 Electron motion through crystals
7.11 Almost Classical Motion
7.11.1 Motion through free space
7.11.2 Accelerated motion
7.11.3 Decelerated motion
7.11.4 The harmonic oscillator
7.12 Scattering
7.12.1 Partial reflection
7.12.2 Tunneling
7.13 Reflection and Transmission Coefficients
8. The Meaning of Quantum Mechanics
8.1 Schrödinger’s Cat
8.2 Instantaneous Interactions
8.3 Global Symmetrization
8.4 A story by Wheeler
8.5 Failure of the Schrödinger Equation?
8.6 The Many-Worlds Interpretation
8.7 The Arrow of Time
9. Numerical Procedures
9.1 The Variational Method
9.1.1 Basic variational statement
9.1.2 Differential form of the statement
9.1.3 Using Lagrangian multipliers
9.2 The Born-Oppenheimer Approximation
9.2.1 The Hamiltonian
9.2.2 Basic Born-Oppenheimer approximation
9.2.3 Going one better
9.3 The Hartree-Fock Approximation
9.3.1 Wave function approximation
9.3.2 The Hamiltonian
9.3.3 The expectation value of energy
9.3.4 The canonical Hartree-Fock equations
9.3.5 Additional points
9.3.5.1 Meaning of the orbital energies
9.3.5.2 Asymptotic behavior
9.3.5.3 Hartree-Fock limit
9.3.5.4 Correlation energy
9.3.5.5 Configuration interaction
10. Solids
10.1 Molecular Solids
10.2 Ionic Solids
10.3 Metals
10.3.1 Lithium
10.3.2 One-dimensional crystals
10.3.3 Wave functions of one-dimensional crystals
10.3.4 Analysis of the wave functions
10.3.5 Floquet (Bloch) theory
10.3.6 Fourier analysis
10.3.7 The reciprocal lattice
10.3.8 The energy levels
10.3.9 Merging and splitting bands
10.3.10 Three-dimensional metals
10.4 Covalent Materials
10.5 Free-Electron Gas
10.5.1 Lattice for the free electrons
10.5.2 Occupied states and Brillouin zones
10.6 Nearly-Free Electrons
10.6.1 Energy changes due to a weak lattice potential
10.6.2 Discussion of the energy changes
10.7 Additional Points
10.7.1 About ferromagnetism
10.7.2 X-ray diffraction
11. Basic and Quantum Thermodynamics
11.1 Temperature
11.2 Single-Particle versus System States
11.3 How Many System Eigenfunctions?
11.4 Particle-Energy Distribution Functions
11.5 The Canonical Probability Distribution
11.6 Low Temperature Behavior
11.7 The Basic Thermodynamic Variables
11.8 Intro to the Second Law
11.9 The Reversible Ideal
11.10 Entropy
11.11 The Big Lie of Distinguishable Particles
11.12 The New Variables
11.13 Microscopic Meaning of the Variables
11.14 Application to Particles in a Box
11.14.1 Bose-Einstein condensation
11.14.2 Fermions at low temperatures
11.14.3 A generalized ideal gas law
11.14.4 The ideal gas
11.14.5 Blackbody radiation
11.14.6 The Debye model
11.15 Specific Heats
12. Angular momentum
12.1 Introduction
12.2 The fundamental commutation relations
12.3 Ladders
12.4 Possible values of angular momentum
12.5 A warning about angular momentum
12.6 Triplet and singlet states
12.7 Clebsch-Gordan coefficients
12.8 Some important results
12.9 Momentum of partially filled shells
12.10 Pauli spin matrices
12.11 General spin matrices
12.12 The Relativistic Dirac Equation
13. Electromagnetism
13.1 The Electromagnetic Hamiltonian
13.2 Maxwell’s Equations
13.3 Example Static Electromagnetic Fields
13.3.1 Point charge at the origin
13.3.2 Dipoles
13.3.3 Arbitrary charge distributions
13.3.4 Solution of the Poisson equation
13.3.5 Currents
13.3.6 Principle of the electric motor
13.4 Particles in Magnetic Fields
13.5 Stern-Gerlach Apparatus
13.6 Nuclear Magnetic Resonance
13.6.1 Description of the method
13.6.2 The Hamiltonian
13.6.3 The unperturbed system
13.6.4 Effect of the perturbation
14. Nuclei [Unfinished Draft]
14.1 Fundamental Concepts
14.2 Draft: The Simplest Nuclei
14.2.1 Draft: The proton
14.2.2 Draft: The neutron
14.2.3 Draft: The deuteron
14.2.4 Draft: Property summary
14.3 Draft: Overview of Nuclei
14.4 Draft: Magic numbers
14.5 Draft: Radioactivity
14.5.1 Draft: Half-life and decay rate
14.5.2 Draft: More than one decay process
14.5.3 Draft: Other definitions
14.6 Draft: Mass and energy
14.7 Draft: Binding energy
14.8 Draft: Nucleon separation energies
14.9 Draft: Modeling the Deuteron
14.10 Draft: Liquid drop model
14.10.1 Draft: Nuclear radius
14.10.2 Draft: von Weizsäcker formula
14.10.3 Draft: Explanation of the formula
14.10.4 Draft: Accuracy of the formula
14.11 Draft: Alpha Decay
14.11.1 Draft: Decay mechanism
14.11.2 Draft: Comparison with data
14.11.3 Draft: Forbidden decays
14.11.4 Draft: Why alpha decay?
14.12 Draft: Shell model
14.12.1 Draft: Average potential
14.12.2 Draft: Spin-orbit interaction
14.12.3 Draft: Example occupation levels
14.12.4 Draft: Shell model with pairing
14.12.5 Draft: Configuration mixing
14.12.6 Draft: Shell model failures
14.13 Draft: Collective Structure
14.13.1 Draft: Classical liquid drop
14.13.2 Draft: Nuclear vibrations
14.13.3 Draft: Nonspherical nuclei
14.13.4 Draft: Rotational bands
14.13.4.1 Draft: Basic notions in nuclear rotation
14.13.4.2 Draft: Basic rotational bands
14.13.4.3 Draft: Bands with intrinsic spin one-half
14.13.4.4 Draft: Bands with intrinsic spin zero
14.13.4.5 Draft: Even-even nuclei
14.13.4.6 Draft: Nonaxial nuclei
14.14 Draft: Fission
14.14.1 Draft: Basic concepts
14.14.2 Draft: Some basic features
14.15 Draft: Spin Data
14.15.1 Draft: Even-even nuclei
14.15.2 Draft: Odd mass number nuclei
14.15.3 Draft: Odd-odd nuclei
14.16 Draft: Parity Data
14.16.1 Draft: Even-even nuclei
14.16.2 Draft: Odd mass number nuclei
14.16.3 Draft: Odd-odd nuclei
14.16.4 Draft: Parity Summary
14.17 Draft: Electromagnetic Moments
14.17.1 Draft: Classical description
14.17.1.1 Draft: Magnetic dipole moment
14.17.1.2 Draft: Electric quadrupole moment
14.17.2 Draft: Quantum description
14.17.2.1 Draft: Magnetic dipole moment
14.17.2.2 Draft: Electric quadrupole moment
14.17.2.3 Draft: Shell model values
14.17.2.4 Draft: Values for deformed nuclei
14.17.3 Draft: Magnetic moment data
14.17.4 Draft: Quadrupole moment data
14.18 Draft: Isospin
14.18.1 Draft: Basic ideas
14.18.2 Draft: Heavier nuclei
14.18.3 Draft: Additional points
14.18.4 Draft: Why does this work?
14.19 Draft: Beta decay
14.19.1 Draft: Introduction
14.19.2 Draft: Energetics Data
14.19.3 Draft: Beta decay and magic numbers
14.19.4 Draft: Von Weizsäcker approximation
14.19.5 Draft: Kinetic Energies
14.19.6 Draft: Forbidden decays
14.19.6.1 Draft: Allowed decays
14.19.6.2 Draft: Forbidden decays allowed
14.19.6.3 Draft: The energy effect
14.19.7 Draft: Data and Fermi theory
14.19.8 Draft: Parity violation
14.20 Draft: Gamma Decay
14.20.1 Draft: Energetics
14.20.2 Draft: Forbidden decays
14.20.3 Draft: Isomers
14.20.4 Draft: Weisskopf estimates
14.20.5 Draft: Comparison with data
14.20.6 Draft: Internal conversion
A. Addenda
A.1 Classical Lagrangian mechanics
A.1.1 Introduction
A.1.2 Generalized coordinates
A.1.3 Lagrangian equations of motion
A.1.4 Hamiltonian dynamics
A.1.5 Fields
A.2 An example of variational calculus
A.3 Galilean transformation
A.4 More on index notation
A.5 The reduced mass
A.6 Constant spherical potentials
A.6.1 The eigenvalue problem
A.6.2 The eigenfunctions
A.6.3 About free space solutions
A.7 Accuracy of the variational method
A.8 Positive ground state wave function
A.9 Wave function symmetries
A.10 Spin inner product
A.11 Thermoelectric effects
A.11.1 Peltier and Seebeck coefficient ballparks
A.11.2 Figure of merit
A.11.3 Physical Seebeck mechanism
A.11.4 Full thermoelectric equations
A.11.5 Charge locations in thermoelectrics
A.11.6 Kelvin relationships
A.12 Heisenberg picture
A.13 Integral Schrödinger equation
A.14 The Klein-Gordon equation
A.15 Quantum Field Theory in a Nanoshell
A.15.1 Occupation numbers
A.15.2 Creation and annihilation operators
A.15.3 The caHermitians
A.15.4 Recasting a Hamiltonian as a quantum field one
A.15.5 The harmonic oscillator as a boson system
A.15.6 Canonical (second) quantization
A.15.7 Spin as a fermion system
A.15.8 More single particle states
A.15.9 Field operators
A.15.10 Nonrelativistic quantum field theory
A.16 The adiabatic theorem
A.17 The virial theorem
A.18 The energy-time uncertainty relationship
A.19 Conservation Laws and Symmetries
A.19.1 An example symmetry transformation
A.19.2 Physical description of a symmetry
A.19.3 Derivation of the conservation law
A.19.4 Other symmetries
A.19.5 A gauge symmetry and conservation of charge
A.19.6 Reservations about time shift symmetry
A.20 Angular momentum of vector particles
A.21 Photon type 2 wave function
A.21.1 The wave function
A.21.2 Simplifying the wave function
A.21.3 Photon spin
A.21.4 Energy eigenstates
A.21.5 Normalization of the wave function
A.21.6 States of definite linear momentum
A.21.7 States of definite angular momentum
A.22 Forces by particle exchange
A.22.1 Classical selectostatics
A.22.2 Classical selectodynamics
A.22.3 Quantum selectostatics
A.22.4 Poincaré and Einstein try to save the universe
A.22.5 Lorenz saves the universe
A.22.6 Gupta-Bleuler condition
A.22.7 The conventional Lagrangian
A.22.8 Quantization following Fermi
A.22.9 The Coulomb potential and the speed of light
A.23 Quantization of radiation
A.23.1 Properties of classical electromagnetic fields
A.23.2 Photon wave functions
A.23.3 The electromagnetic operators
A.23.4 Properties of the observable electromagnetic field
A.24 Quantum spontaneous emission
A.25 Multipole transitions
A.25.1 Approximate Hamiltonian
A.25.2 Approximate multipole matrix elements
A.25.3 Corrected multipole matrix elements
A.25.4 Matrix element ballparks
A.25.5 Selection rules
A.25.6 Ballpark decay rates
A.25.7 Wave functions of definite angular momentum
A.25.8 Weisskopf and Moszkowski estimates
A.25.9 Errors in other sources
A.26 Fourier inversion theorem and Parseval
A.27 Details of the animations
A.28 WKB Theory of Nearly Classical Motion
A.28.1 Solution wkb-a
A.28.2 Solution wkb-b
A.29 WKB solution near the turning points
A.30 Three-dimensional scattering
A.30.1 Partial wave analysis
A.30.2 Partial wave amplitude
A.30.3 The Born approximation
A.31 The Born series
A.32 The evolution of probability
A.33 Explanation of the London forces
A.34 Explanation of Hund’s first rule
A.35 The third law
A.36 Alternate Dirac equations
A.37 Maxwell’s wave equations
A.38 Perturbation Theory
A.38.1 Basic perturbation theory
A.38.2 Ionization energy of helium
A.38.3 Degenerate perturbation theory
A.38.4 The Zeeman effect
A.38.5 The Stark effect
A.39 The relativistic hydrogen atom
A.39.1 Introduction
A.39.2 Fine structure
A.39.3 Weak and intermediate Zeeman effect
A.39.4 Lamb shift
A.39.5 Hyperfine splitting
A.40 Deuteron wave function
A.41 Deuteron model
A.41.1 The model
A.41.2 The repulsive core
A.41.3 Spin dependence
A.41.4 Noncentral force
A.41.5 Spin-orbit interaction
A.42 Nuclear forces
A.42.1 Basic Yukawa potential
A.42.2 OPEP potential
A.42.3 Explanation of the OPEP potential
A.42.4 Multiple pion exchange and such
A.43 Classical vibrating drop
A.43.1 Basic definitions
A.43.2 Kinetic energy
A.43.3 Energy due to surface tension
A.43.4 Energy due to Coulomb repulsion
A.43.5 Frequency of vibration
A.44 Relativistic neutrinos
A.45 Fermi theory
A.45.1 Form of the wave function
A.45.2 Source of the decay
A.45.3 Allowed or forbidden
A.45.4 The nuclear operator
A.45.5 Fermi’s golden rule
A.45.6 Mopping up
A.45.7 Electron capture
D. Derivations
D.1 Generic vector identities
D.2 Some Green’s functions
D.2.1 The Poisson equation
D.2.2 The screened Poisson equation
D.3 Lagrangian mechanics
D.3.1 Lagrangian equations of motion
D.3.2 Hamiltonian dynamics
D.3.3 Fields
D.4 Lorentz transformation derivation
D.5 Lorentz group property derivation
D.6 Lorentz force derivation
D.7 Derivation of the Euler formula
D.8 Completeness of Fourier modes
D.9 Momentum operators are Hermitian
D.10 The curl is Hermitian
D.11 Extension to three-dimensional solutions
D.12 The harmonic oscillator solution
D.13 The harmonic oscillator and uncertainty
D.14 The spherical harmonics
D.14.1 Derivation from the eigenvalue problem
D.14.2 Parity
D.14.3 Solutions of the Laplace equation
D.14.4 Orthogonal integrals
D.14.5 Another way to find the spherical harmonics
D.14.6 Still another way to find them
D.15 The hydrogen radial wave functions
D.16 Constant spherical potentials derivations
D.16.1 The eigenfunctions
D.16.2 The Rayleigh formula
D.17 Inner product for the expectation value
D.18 Eigenfunctions of commuting operators
D.19 The generalized uncertainty relationship
D.20 Derivation of the commutator rules
D.21 Solution of the hydrogen molecular ion
D.22 Unique ground state wave function
D.23 Solution of the hydrogen molecule
D.24 Hydrogen molecule ground state and spin
D.25 Number of boson states
D.26 Density of states
D.27 Radiation from a hole
D.28 Kirchhoff’s law
D.29 The thermionic emission equation
D.30 Number of conduction band electrons
D.31 Integral Schrödinger equation
D.32 Integral conservation laws
D.33 Quantum field derivations
D.34 The adiabatic theorem
D.35 The evolution of expectation values
D.36 Photon wave function derivations
D.36.1 Rewriting the energy integral
D.36.2 Angular momentum states
D.36.2.1 About the scalar modes
D.36.2.2 Basic observations and eigenvalue problem
D.36.2.3 Spherical form and net angular momentum
D.36.2.4 Orthogonality and normalization
D.36.2.5 Completeness
D.36.2.6 Density of states
D.36.2.7 Parity
D.36.2.8 Orbital angular momentum of the states
D.37 Forces by particle exchange derivations
D.37.1 Classical energy minimization
D.37.2 Quantum energy minimization
D.37.3 Rewriting the Lagrangian
D.37.4 Coulomb potential energy
D.38 Time-dependent perturbation theory
D.39 Selection rules
D.40 Quantization of radiation derivations
D.41 Derivation of the Einstein B coefficients
D.42 Derivation of the Einstein A coefficients
D.43 Multipole derivations
D.43.1 Matrix element for linear momentum modes
D.43.2 Matrix element for angular momentum modes
D.43.3 Weisskopf and Moszkowski estimates
D.44 Derivation of group velocity
D.45 Motion through crystals
D.45.1 Propagation speed
D.45.2 Motion under an external force
D.45.3 Free-electron gas with constant electric field
D.46 Derivation of the WKB approximation
D.47 Born differential cross section
D.48 About Lagrangian multipliers
D.49 The generalized variational principle
D.50 Spin degeneracy
D.51 Born-Oppenheimer nuclear motion
D.52 Simplification of the Hartree-Fock energy
D.53 Integral constraints
D.54 Derivation of the Hartree-Fock equations
D.55 Why the Fock operator is Hermitian
D.56 Number of system eigenfunctions
D.57 The particle energy distributions
D.58 The canonical probability distribution
D.59 Analysis of the ideal gas Carnot cycle
D.60 Checks on the expression for entropy
D.61 Chemical potential in the distributions
D.62 Fermi-Dirac integrals at low temperature
D.63 Angular momentum uncertainty
D.64 Spherical harmonics by ladder operators
D.65 How to make Clebsch-Gordan tables
D.66 The triangle inequality
D.67 Momentum of shells
D.68 Awkward questions about spin
D.69 More awkwardness about spin
D.70 Emergence of spin from relativity
D.71 Electromagnetic commutators
D.72 Various electrostatic derivations.
D.72.1 Existence of a potential
D.72.2 The Laplace equation
D.72.3 Egg-shaped dipole field lines
D.72.4 Ideal charge dipole delta function
D.72.5 Integrals of the current density
D.72.6 Lorentz forces on a current distribution
D.72.7 Field of a current dipole
D.72.8 Biot-Savart law
D.73 Orbital motion in a magnetic field
D.74 Electron spin in a magnetic field
D.75 Solving the NMR equations
D.76 Harmonic oscillator revisited
D.77 Impenetrable spherical shell
D.78 Shell model quadrupole moment
D.79 Derivation of perturbation theory
D.80 Hydrogen ground state Stark effect
D.81 Dirac fine structure Hamiltonian
D.82 Classical spin-orbit derivation
D.83 Expectation powers of r for hydrogen
D.84 Band gap explanation derivations
N. Notes
N.1 Why this book?
N.2 History and wish list
N.3 Nature and real eigenvalues
N.4 Are Hermitian operators really like that?
N.5 Why boundary conditions are tricky
N.6 Is the variational approximation best?
N.7 Shielding approximation limitations
N.8 Why the s states have the least energy
N.9 Explanation of the band gaps
N.10 A less fishy story
N.11 Better description of two-state systems
N.12 Second quantization in other books
N.13 Combining angular momentum factors
N.14 The electric multipole problem
N.15 A tenth of a googol in universes
N.16 A single Slater determinant is not exact
N.17 Generalized orbitals
N.18 Correlation energy
N.19 Ambiguities in electron affinity
N.20 Why Floquet theory should be called so
N.21 Superfluidity versus BEC
N.22 The mechanism of ferromagnetism
N.23 Fundamental assumption of statistics
N.24 A problem if the energy is given
N.25 The recipe of life
N.26 Physics of the fundamental commutators
N.27 Magnitude of components of vectors
N.28 Adding angular momentum components
N.29 Clebsch-Gordan tables are bidirectional
N.30 Machine language Clebsch-Gordan tables
N.31 Existence of magnetic monopoles
N.32 More on Maxwell’s third law
N.33 Setting the record straight on alignment
N.34 NuDat 2 data selection
N.35 Auger discovery
N.36 Draft: Cage-of-Faraday proposal
Web Pages
References
Subsections
1
. Special Relativity [Draft]
1
.
1
Overview of Relativity
1
.
1
.
1
A note on the history of the theory
1
.
1
.
2
The mass-energy relation
1
.
1
.
3
The universal speed of light
1
.
1
.
4
Disagreements about space and time
1
.
2
The Lorentz Transformation
1
.
2
.
1
The transformation formulae
1
.
2
.
2
Proper time and distance
1
.
2
.
3
Subluminal and superluminal effects
1
.
2
.
4
Four-vectors
1
.
2
.
5
Index notation
1
.
2
.
6
Group property
1
.
3
Relativistic Mechanics
1
.
3
.
1
Intro to relativistic mechanics
1
.
3
.
2
Lagrangian mechanics
2
. Mathematical Prerequisites
2
.
1
Complex Numbers
2
.
1
.
1
Solution mathcplx-a
2
.
1
.
2
Solution mathcplx-b
2
.
1
.
3
Solution mathcplx-c
2
.
1
.
4
Solution mathcplx-d
2
.
1
.
5
Solution mathcplx-e
2
.
1
.
6
Solution mathcplx-f
2
.
1
.
7
Solution mathcplx-g
2
.
1
.
8
Solution mathcplx-h
2
.
2
Functions as Vectors
2
.
2
.
1
Solution funcvec-a
2
.
2
.
2
Solution funcvec-b
2
.
3
The Dot, oops, INNER Product
2
.
3
.
1
Solution dot-a
2
.
3
.
2
Solution dot-b
2
.
3
.
3
Solution dot-c
2
.
3
.
4
Solution dot-d
2
.
3
.
5
Solution dot-e
2
.
3
.
6
Solution dot-f
2
.
3
.
7
Solution dot-g
2
.
4
Operators
2
.
4
.
1
Solution mathops-a
2
.
4
.
2
Solution mathops-b
2
.
4
.
3
Solution mathops-c
2
.
4
.
4
Solution mathops-d
2
.
5
Eigenvalue Problems
2
.
5
.
1
Solution eigvals-a
2
.
5
.
2
Solution eigvals-b
2
.
5
.
3
Solution eigvals-c
2
.
6
Hermitian Operators
2
.
6
.
1
Solution herm-a
2
.
6
.
2
Solution herm-b
2
.
6
.
3
Solution herm-c
2
.
6
.
4
Solution herm-d
2
.
6
.
5
Solution herm-e
2
.
6
.
6
Solution herm-f
2
.
6
.
7
Solution herm-g
2
.
6
.
8
Solution herm-h
2
.
6
.
9
Solution herm-i
2
.
7
Additional Points
2
.
7
.
1
Dirac notation
2
.
7
.
2
Additional independent variables
3
. Basic Ideas of Quantum Mechanics
3
.
1
The Revised Picture of Nature
3
.
2
The Heisenberg Uncertainty Principle
3
.
3
The Operators of Quantum Mechanics
3
.
4
The Orthodox Statistical Interpretation
3
.
4
.
1
Only eigenvalues
3
.
4
.
2
Statistical selection
3
.
5
A Particle Confined Inside a Pipe
3
.
5
.
1
The physical system
3
.
5
.
2
Mathematical notations
3
.
5
.
3
The Hamiltonian
3
.
5
.
4
The Hamiltonian eigenvalue problem
3
.
5
.
5
All solutions of the eigenvalue problem
3
.
5
.
5
.
1
Solution piped-a
3
.
5
.
5
.
2
Solution piped-b
3
.
5
.
6
Discussion of the energy values
3
.
5
.
6
.
1
Solution pipee-a
3
.
5
.
6
.
2
Solution pipee-b
3
.
5
.
6
.
3
Solution pipee-c
3
.
5
.
7
Discussion of the eigenfunctions
3
.
5
.
7
.
1
Solution pipef-a
3
.
5
.
7
.
2
Solution pipef-b
3
.
5
.
7
.
3
Solution pipef-c
3
.
5
.
8
Three-dimensional solution
3
.
5
.
8
.
1
Solution pipeg-a
3
.
5
.
8
.
2
Solution pipeg-b
3
.
5
.
8
.
3
Solution pipeg-c
3
.
5
.
9
Quantum confinement
4
. Single-Particle Systems
4
.
1
The Harmonic Oscillator
4
.
1
.
1
The Hamiltonian
4
.
1
.
2
Solution using separation of variables
4
.
1
.
2
.
1
Solution harmb-a
4
.
1
.
2
.
2
Solution harmb-b
4
.
1
.
2
.
3
Solution harmb-c
4
.
1
.
2
.
4
Solution harmb-d
4
.
1
.
3
Discussion of the eigenvalues
4
.
1
.
3
.
1
Solution harmc-a
4
.
1
.
3
.
2
Solution harmc-b
4
.
1
.
4
Discussion of the eigenfunctions
4
.
1
.
4
.
1
Solution harmd-a
4
.
1
.
4
.
2
Solution harmd-b
4
.
1
.
4
.
3
Solution harmd-c
4
.
1
.
5
Degeneracy
4
.
1
.
5
.
1
Solution harme-a
4
.
1
.
6
Noneigenstates
4
.
2
Angular Momentum
4
.
2
.
1
Definition of angular momentum
4
.
2
.
2
Angular momentum in an arbitrary direction
4
.
2
.
2
.
1
Solution angub-a
4
.
2
.
2
.
2
Solution angub-b
4
.
2
.
2
.
3
Solution angub-c
4
.
2
.
3
Square angular momentum
4
.
2
.
3
.
1
Solution anguc-a
4
.
2
.
3
.
2
Solution anguc-b
4
.
2
.
3
.
3
Solution anguc-c
4
.
2
.
4
Angular momentum uncertainty
4
.
3
The Hydrogen Atom
4
.
3
.
1
The Hamiltonian
4
.
3
.
2
Solution using separation of variables
4
.
3
.
2
.
1
Solution hydb-a
4
.
3
.
2
.
2
Solution hydb-b
4
.
3
.
2
.
3
Solution hydb-c
4
.
3
.
3
Discussion of the eigenvalues
4
.
3
.
3
.
1
Solution hydc-a
4
.
3
.
3
.
2
Solution hydc-b
4
.
3
.
3
.
3
Solution hydc-c
4
.
3
.
3
.
4
Solution hydc-d
4
.
3
.
4
Discussion of the eigenfunctions
4
.
3
.
4
.
1
Solution hydd-a
4
.
3
.
4
.
2
Solution hydd-b
4
.
3
.
4
.
3
Solution hydd-c
4
.
4
Expectation Value and Standard Deviation
4
.
4
.
1
Statistics of a die
4
.
4
.
1
.
1
Solution esda-a
4
.
4
.
1
.
2
Solution esda-b
4
.
4
.
1
.
3
Solution esda-c
4
.
4
.
1
.
4
Solution esda-d
4
.
4
.
2
Statistics of quantum operators
4
.
4
.
2
.
1
Solution esdb-a
4
.
4
.
2
.
2
Solution esdb-b
4
.
4
.
3
Simplified expressions
4
.
4
.
3
.
1
Solution esdb2-a
4
.
4
.
3
.
2
Solution esdb2-b
4
.
4
.
4
Some examples
4
.
5
The Commutator
4
.
5
.
1
Commuting operators
4
.
5
.
1
.
1
Solution commutea-a
4
.
5
.
2
Noncommuting operators and their commutator
4
.
5
.
3
The Heisenberg uncertainty relationship
4
.
5
.
3
.
1
Solution commutec-a
4
.
5
.
4
Commutator reference
4
.
6
The Hydrogen Molecular Ion
4
.
6
.
1
The Hamiltonian
4
.
6
.
2
Energy when fully dissociated
4
.
6
.
3
Energy when closer together
4
.
6
.
4
States that share the electron
4
.
6
.
5
Comparative energies of the states
4
.
6
.
6
Variational approximation of the ground state
4
.
6
.
6
.
1
Solution hione-a
4
.
6
.
7
Comparison with the exact ground state
5
. Multiple-Particle Systems
5
.
1
Wave Function for Multiple Particles
5
.
1
.
1
Solution complex-a
5
.
1
.
2
Solution complex-b
5
.
2
The Hydrogen Molecule
5
.
2
.
1
The Hamiltonian
5
.
2
.
1
.
1
Solution hmola-a
5
.
2
.
1
.
2
Solution hmola-b
5
.
2
.
2
Initial approximation to the lowest energy state
5
.
2
.
2
.
1
Solution hmolb-a
5
.
2
.
2
.
2
Solution hmolb-b
5
.
2
.
3
The probability density
5
.
2
.
3
.
1
Solution hmolc-a
5
.
2
.
4
States that share the electrons
5
.
2
.
4
.
1
Solution hmold-a
5
.
2
.
4
.
2
Solution hmold-b
5
.
2
.
5
Variational approximation of the ground state
5
.
2
.
6
Comparison with the exact ground state
5
.
3
Two-State Systems
5
.
3
.
1
Solution 2state-a
5
.
3
.
2
Solution 2state-b
5
.
4
Spin
5
.
4
.
1
Solution spin-a
5
.
4
.
2
Solution spin-b
5
.
5
Multiple-Particle Systems Including Spin
5
.
5
.
1
Wave function for a single particle with spin
5
.
5
.
1
.
1
Solution complexsa-a
5
.
5
.
2
Inner products including spin
5
.
5
.
2
.
1
Solution complexsai-a
5
.
5
.
2
.
2
Solution complexsai-b
5
.
5
.
3
Commutators including spin
5
.
5
.
3
.
1
Solution complexsac-a
5
.
5
.
4
Wave function for multiple particles with spin
5
.
5
.
4
.
1
Solution complexsb-a
5
.
5
.
4
.
2
Solution complexsb-b
5
.
5
.
5
Example: the hydrogen molecule
5
.
5
.
5
.
1
Solution complexsc-a
5
.
5
.
6
Triplet and singlet states
5
.
5
.
6
.
1
Solution complexse-a
5
.
6
Identical Particles
5
.
6
.
1
Solution ident-a
5
.
6
.
2
Solution ident-b
5
.
7
Ways to Symmetrize the Wave Function
5
.
7
.
1
Solution symways-a
5
.
7
.
2
Solution symways-b
5
.
8
Matrix Formulation
5
.
8
.
1
Solution matfor-a
5
.
8
.
2
Solution matfor-b
5
.
9
Heavier Atoms
5
.
9
.
1
The Hamiltonian eigenvalue problem
5
.
9
.
2
Approximate solution using separation of variables
5
.
9
.
3
Hydrogen and helium
5
.
9
.
4
Lithium to neon
5
.
9
.
5
Sodium to argon
5
.
9
.
6
Potassium to krypton
5
.
9
.
7
Full periodic table
5
.
10
Pauli Repulsion
5
.
11
Chemical Bonds
5
.
11
.
1
Covalent sigma bonds
5
.
11
.
2
Covalent pi bonds
5
.
11
.
3
Polar covalent bonds and hydrogen bonds
5
.
11
.
4
Promotion and hybridization
5
.
11
.
5
Ionic bonds
5
.
11
.
6
Limitations of valence bond theory
6
. Macroscopic Systems
6
.
1
Intro to Particles in a Box
6
.
2
The Single-Particle States
6
.
3
Density of States
6
.
4
Ground State of a System of Bosons
6
.
5
About Temperature
6
.
6
Bose-Einstein Condensation
6
.
6
.
1
Rough explanation of the condensation
6
.
7
Bose-Einstein Distribution
6
.
8
Blackbody Radiation
6
.
9
Ground State of a System of Electrons
6
.
10
Fermi Energy of the Free-Electron Gas
6
.
11
Degeneracy Pressure
6
.
12
Confinement and the DOS
6
.
13
Fermi-Dirac Distribution
6
.
14
Maxwell-Boltzmann Distribution
6
.
15
Thermionic Emission
6
.
16
Chemical Potential and Diffusion
6
.
17
Intro to the Periodic Box
6
.
18
Periodic Single-Particle States
6
.
19
DOS for a Periodic Box
6
.
20
Intro to Electrical Conduction
6
.
21
Intro to Band Structure
6
.
21
.
1
Metals and insulators
6
.
21
.
2
Typical metals and insulators
6
.
21
.
3
Semiconductors
6
.
21
.
4
Semimetals
6
.
21
.
5
Electronic heat conduction
6
.
21
.
6
Ionic conductivity
6
.
22
Electrons in Crystals
6
.
22
.
1
Bloch waves
6
.
22
.
2
Example spectra
6
.
22
.
3
Effective mass
6
.
22
.
4
Crystal momentum
6
.
22
.
5
Three-dimensional crystals
6
.
23
Semiconductors
6
.
24
The
P-N
Junction
6
.
25
The Transistor
6
.
26
Zener and Avalanche Diodes
6
.
27
Optical Applications
6
.
27
.
1
Atomic spectra
6
.
27
.
2
Spectra of solids
6
.
27
.
3
Band gap effects
6
.
27
.
4
Effects of crystal imperfections
6
.
27
.
5
Photoconductivity
6
.
27
.
6
Photovoltaic cells
6
.
27
.
7
Light-emitting diodes
6
.
28
Thermoelectric Applications
6
.
28
.
1
Peltier effect
6
.
28
.
2
Seebeck effect
6
.
28
.
3
Thomson effect
7
. Time Evolution
7
.
1
The Schrödinger Equation
7
.
1
.
1
The equation
7
.
1
.
2
Solution of the equation
7
.
1
.
2
.
1
Solution schrodsol-a
7
.
1
.
2
.
2
Solution schrodsol-b
7
.
1
.
2
.
3
Solution schrodsol-c
7
.
1
.
3
Energy conservation
7
.
1
.
4
Stationary states
7
.
1
.
5
The adiabatic approximation
7
.
2
Time Variation of Expectation Values
7
.
2
.
1
Newtonian motion
7
.
2
.
2
Energy-time uncertainty relation
7
.
3
Conservation Laws and Symmetries
7
.
4
Conservation Laws in Emission
7
.
4
.
1
Conservation of energy
7
.
4
.
2
Combining angular momenta and parities
7
.
4
.
3
Transition types and their photons
7
.
4
.
4
Selection rules
7
.
5
Symmetric Two-State Systems
7
.
5
.
1
A graphical example
7
.
5
.
2
Particle exchange and forces
7
.
5
.
3
Spontaneous emission
7
.
6
Asymmetric Two-State Systems
7
.
6
.
1
Spontaneous emission revisited
7
.
7
Absorption and Stimulated Emission
7
.
7
.
1
The Hamiltonian
7
.
7
.
2
The two-state model
7
.
8
General Interaction with Radiation
7
.
9
Position and Linear Momentum
7
.
9
.
1
The position eigenfunction
7
.
9
.
2
The linear momentum eigenfunction
7
.
10
Wave Packets
7
.
10
.
1
Solution of the Schrödinger equation.
7
.
10
.
2
Component wave solutions
7
.
10
.
3
Wave packets
7
.
10
.
4
Group velocity
7
.
10
.
5
Electron motion through crystals
7
.
11
Almost Classical Motion
7
.
11
.
1
Motion through free space
7
.
11
.
2
Accelerated motion
7
.
11
.
3
Decelerated motion
7
.
11
.
4
The harmonic oscillator
7
.
12
Scattering
7
.
12
.
1
Partial reflection
7
.
12
.
2
Tunneling
7
.
13
Reflection and Transmission Coefficients
8
. The Meaning of Quantum Mechanics
8
.
1
Schrödinger’s Cat
8
.
2
Instantaneous Interactions
8
.
3
Global Symmetrization
8
.
4
A story by Wheeler
8
.
5
Failure of the Schrödinger Equation?
8
.
6
The Many-Worlds Interpretation
8
.
7
The Arrow of Time
9
. Numerical Procedures
9
.
1
The Variational Method
9
.
1
.
1
Basic variational statement
9
.
1
.
2
Differential form of the statement
9
.
1
.
3
Using Lagrangian multipliers
9
.
2
The Born-Oppenheimer Approximation
9
.
2
.
1
The Hamiltonian
9
.
2
.
2
Basic Born-Oppenheimer approximation
9
.
2
.
3
Going one better
9
.
3
The Hartree-Fock Approximation
9
.
3
.
1
Wave function approximation
9
.
3
.
2
The Hamiltonian
9
.
3
.
3
The expectation value of energy
9
.
3
.
4
The canonical Hartree-Fock equations
9
.
3
.
5
Additional points
9
.
3
.
5
.
1
Meaning of the orbital energies
9
.
3
.
5
.
2
Asymptotic behavior
9
.
3
.
5
.
3
Hartree-Fock limit
9
.
3
.
5
.
4
Correlation energy
9
.
3
.
5
.
5
Configuration interaction
10
. Solids
10
.
1
Molecular Solids
10
.
2
Ionic Solids
10
.
3
Metals
10
.
3
.
1
Lithium
10
.
3
.
2
One-dimensional crystals
10
.
3
.
3
Wave functions of one-dimensional crystals
10
.
3
.
4
Analysis of the wave functions
10
.
3
.
5
Floquet (Bloch) theory
10
.
3
.
6
Fourier analysis
10
.
3
.
7
The reciprocal lattice
10
.
3
.
8
The energy levels
10
.
3
.
9
Merging and splitting bands
10
.
3
.
10
Three-dimensional metals
10
.
4
Covalent Materials
10
.
5
Free-Electron Gas
10
.
5
.
1
Lattice for the free electrons
10
.
5
.
2
Occupied states and Brillouin zones
10
.
6
Nearly-Free Electrons
10
.
6
.
1
Energy changes due to a weak lattice potential
10
.
6
.
2
Discussion of the energy changes
10
.
7
Additional Points
10
.
7
.
1
About ferromagnetism
10
.
7
.
2
X-ray diffraction
11
. Basic and Quantum Thermodynamics
11
.
1
Temperature
11
.
2
Single-Particle versus System States
11
.
3
How Many System Eigenfunctions?
11
.
4
Particle-Energy Distribution Functions
11
.
5
The Canonical Probability Distribution
11
.
6
Low Temperature Behavior
11
.
7
The Basic Thermodynamic Variables
11
.
8
Intro to the Second Law
11
.
9
The Reversible Ideal
11
.
10
Entropy
11
.
11
The Big Lie of Distinguishable Particles
11
.
12
The New Variables
11
.
13
Microscopic Meaning of the Variables
11
.
14
Application to Particles in a Box
11
.
14
.
1
Bose-Einstein condensation
11
.
14
.
2
Fermions at low temperatures
11
.
14
.
3
A generalized ideal gas law
11
.
14
.
4
The ideal gas
11
.
14
.
5
Blackbody radiation
11
.
14
.
6
The Debye model
11
.
15
Specific Heats
12
. Angular momentum
12
.
1
Introduction
12
.
2
The fundamental commutation relations
12
.
3
Ladders
12
.
4
Possible values of angular momentum
12
.
5
A warning about angular momentum
12
.
6
Triplet and singlet states
12
.
7
Clebsch-Gordan coefficients
12
.
8
Some important results
12
.
9
Momentum of partially filled shells
12
.
10
Pauli spin matrices
12
.
11
General spin matrices
12
.
12
The Relativistic Dirac Equation
13
. Electromagnetism
13
.
1
The Electromagnetic Hamiltonian
13
.
2
Maxwell’s Equations
13
.
3
Example Static Electromagnetic Fields
13
.
3
.
1
Point charge at the origin
13
.
3
.
2
Dipoles
13
.
3
.
3
Arbitrary charge distributions
13
.
3
.
4
Solution of the Poisson equation
13
.
3
.
5
Currents
13
.
3
.
6
Principle of the electric motor
13
.
4
Particles in Magnetic Fields
13
.
5
Stern-Gerlach Apparatus
13
.
6
Nuclear Magnetic Resonance
13
.
6
.
1
Description of the method
13
.
6
.
2
The Hamiltonian
13
.
6
.
3
The unperturbed system
13
.
6
.
4
Effect of the perturbation
14
. Nuclei [Unfinished Draft]
14
.
1
Fundamental Concepts
14
.
2
Draft: The Simplest Nuclei
14
.
2
.
1
Draft: The proton
14
.
2
.
2
Draft: The neutron
14
.
2
.
3
Draft: The deuteron
14
.
2
.
4
Draft: Property summary
14
.
3
Draft: Overview of Nuclei
14
.
4
Draft: Magic numbers
14
.
5
Draft: Radioactivity
14
.
5
.
1
Draft: Half-life and decay rate
14
.
5
.
2
Draft: More than one decay process
14
.
5
.
3
Draft: Other definitions
14
.
6
Draft: Mass and energy
14
.
7
Draft: Binding energy
14
.
8
Draft: Nucleon separation energies
14
.
9
Draft: Modeling the Deuteron
14
.
10
Draft: Liquid drop model
14
.
10
.
1
Draft: Nuclear radius
14
.
10
.
2
Draft: von Weizsäcker formula
14
.
10
.
3
Draft: Explanation of the formula
14
.
10
.
4
Draft: Accuracy of the formula
14
.
11
Draft: Alpha Decay
14
.
11
.
1
Draft: Decay mechanism
14
.
11
.
2
Draft: Comparison with data
14
.
11
.
3
Draft: Forbidden decays
14
.
11
.
4
Draft: Why alpha decay?
14
.
12
Draft: Shell model
14
.
12
.
1
Draft: Average potential
14
.
12
.
2
Draft: Spin-orbit interaction
14
.
12
.
3
Draft: Example occupation levels
14
.
12
.
4
Draft: Shell model with pairing
14
.
12
.
5
Draft: Configuration mixing
14
.
12
.
6
Draft: Shell model failures
14
.
13
Draft: Collective Structure
14
.
13
.
1
Draft: Classical liquid drop
14
.
13
.
2
Draft: Nuclear vibrations
14
.
13
.
3
Draft: Nonspherical nuclei
14
.
13
.
4
Draft: Rotational bands
14
.
13
.
4
.
1
Draft: Basic notions in nuclear rotation
14
.
13
.
4
.
2
Draft: Basic rotational bands
14
.
13
.
4
.
3
Draft: Bands with intrinsic spin one-half
14
.
13
.
4
.
4
Draft: Bands with intrinsic spin zero
14
.
13
.
4
.
5
Draft: Even-even nuclei
14
.
13
.
4
.
6
Draft: Nonaxial nuclei
14
.
14
Draft: Fission
14
.
14
.
1
Draft: Basic concepts
14
.
14
.
2
Draft: Some basic features
14
.
15
Draft: Spin Data
14
.
15
.
1
Draft: Even-even nuclei
14
.
15
.
2
Draft: Odd mass number nuclei
14
.
15
.
3
Draft: Odd-odd nuclei
14
.
16
Draft: Parity Data
14
.
16
.
1
Draft: Even-even nuclei
14
.
16
.
2
Draft: Odd mass number nuclei
14
.
16
.
3
Draft: Odd-odd nuclei
14
.
16
.
4
Draft: Parity Summary
14
.
17
Draft: Electromagnetic Moments
14
.
17
.
1
Draft: Classical description
14
.
17
.
1
.
1
Draft: Magnetic dipole moment
14
.
17
.
1
.
2
Draft: Electric quadrupole moment
14
.
17
.
2
Draft: Quantum description
14
.
17
.
2
.
1
Draft: Magnetic dipole moment
14
.
17
.
2
.
2
Draft: Electric quadrupole moment
14
.
17
.
2
.
3
Draft: Shell model values
14
.
17
.
2
.
4
Draft: Values for deformed nuclei
14
.
17
.
3
Draft: Magnetic moment data
14
.
17
.
4
Draft: Quadrupole moment data
14
.
18
Draft: Isospin
14
.
18
.
1
Draft: Basic ideas
14
.
18
.
2
Draft: Heavier nuclei
14
.
18
.
3
Draft: Additional points
14
.
18
.
4
Draft: Why does this work?
14
.
19
Draft: Beta decay
14
.
19
.
1
Draft: Introduction
14
.
19
.
2
Draft: Energetics Data
14
.
19
.
3
Draft: Beta decay and magic numbers
14
.
19
.
4
Draft: Von Weizsäcker approximation
14
.
19
.
5
Draft: Kinetic Energies
14
.
19
.
6
Draft: Forbidden decays
14
.
19
.
6
.
1
Draft: Allowed decays
14
.
19
.
6
.
2
Draft: Forbidden decays allowed
14
.
19
.
6
.
3
Draft: The energy effect
14
.
19
.
7
Draft: Data and Fermi theory
14
.
19
.
8
Draft: Parity violation
14
.
20
Draft: Gamma Decay
14
.
20
.
1
Draft: Energetics
14
.
20
.
2
Draft: Forbidden decays
14
.
20
.
3
Draft: Isomers
14
.
20
.
4
Draft: Weisskopf estimates
14
.
20
.
5
Draft: Comparison with data
14
.
20
.
6
Draft: Internal conversion
A. Addenda
A.
1
Classical Lagrangian mechanics
A.
1
.
1
Introduction
A.
1
.
2
Generalized coordinates
A.
1
.
3
Lagrangian equations of motion
A.
1
.
4
Hamiltonian dynamics
A.
1
.
5
Fields
A.
2
An example of variational calculus
A.
3
Galilean transformation
A.
4
More on index notation
A.
5
The reduced mass
A.
6
Constant spherical potentials
A.
6
.
1
The eigenvalue problem
A.
6
.
2
The eigenfunctions
A.
6
.
3
About free space solutions
A.
7
Accuracy of the variational method
A.
8
Positive ground state wave function
A.
9
Wave function symmetries
A.
10
Spin inner product
A.
11
Thermoelectric effects
A.
11
.
1
Peltier and Seebeck coefficient ballparks
A.
11
.
2
Figure of merit
A.
11
.
3
Physical Seebeck mechanism
A.
11
.
4
Full thermoelectric equations
A.
11
.
5
Charge locations in thermoelectrics
A.
11
.
6
Kelvin relationships
A.
12
Heisenberg picture
A.
13
Integral Schrödinger equation
A.
14
The Klein-Gordon equation
A.
15
Quantum Field Theory in a Nanoshell
A.
15
.
1
Occupation numbers
A.
15
.
2
Creation and annihilation operators
A.
15
.
3
The caHermitians
A.
15
.
4
Recasting a Hamiltonian as a quantum field one
A.
15
.
5
The harmonic oscillator as a boson system
A.
15
.
6
Canonical (second) quantization
A.
15
.
7
Spin as a fermion system
A.
15
.
8
More single particle states
A.
15
.
9
Field operators
A.
15
.
10
Nonrelativistic quantum field theory
A.
16
The adiabatic theorem
A.
17
The virial theorem
A.
18
The energy-time uncertainty relationship
A.
19
Conservation Laws and Symmetries
A.
19
.
1
An example symmetry transformation
A.
19
.
2
Physical description of a symmetry
A.
19
.
3
Derivation of the conservation law
A.
19
.
4
Other symmetries
A.
19
.
5
A gauge symmetry and conservation of charge
A.
19
.
6
Reservations about time shift symmetry
A.
20
Angular momentum of vector particles
A.
21
Photon type 2 wave function
A.
21
.
1
The wave function
A.
21
.
2
Simplifying the wave function
A.
21
.
3
Photon spin
A.
21
.
4
Energy eigenstates
A.
21
.
5
Normalization of the wave function
A.
21
.
6
States of definite linear momentum
A.
21
.
7
States of definite angular momentum
A.
22
Forces by particle exchange
A.
22
.
1
Classical selectostatics
A.
22
.
2
Classical selectodynamics
A.
22
.
3
Quantum selectostatics
A.
22
.
4
Poincaré and Einstein try to save the universe
A.
22
.
5
Lorenz saves the universe
A.
22
.
6
Gupta-Bleuler condition
A.
22
.
7
The conventional Lagrangian
A.
22
.
8
Quantization following Fermi
A.
22
.
9
The Coulomb potential and the speed of light
A.
23
Quantization of radiation
A.
23
.
1
Properties of classical electromagnetic fields
A.
23
.
2
Photon wave functions
A.
23
.
3
The electromagnetic operators
A.
23
.
4
Properties of the observable electromagnetic field
A.
24
Quantum spontaneous emission
A.
25
Multipole transitions
A.
25
.
1
Approximate Hamiltonian
A.
25
.
2
Approximate multipole matrix elements
A.
25
.
3
Corrected multipole matrix elements
A.
25
.
4
Matrix element ballparks
A.
25
.
5
Selection rules
A.
25
.
6
Ballpark decay rates
A.
25
.
7
Wave functions of definite angular momentum
A.
25
.
8
Weisskopf and Moszkowski estimates
A.
25
.
9
Errors in other sources
A.
26
Fourier inversion theorem and Parseval
A.
27
Details of the animations
A.
28
WKB Theory of Nearly Classical Motion
A.
28
.
1
Solution wkb-a
A.
28
.
2
Solution wkb-b
A.
29
WKB solution near the turning points
A.
30
Three-dimensional scattering
A.
30
.
1
Partial wave analysis
A.
30
.
2
Partial wave amplitude
A.
30
.
3
The Born approximation
A.
31
The Born series
A.
32
The evolution of probability
A.
33
Explanation of the London forces
A.
34
Explanation of Hund’s first rule
A.
35
The third law
A.
36
Alternate Dirac equations
A.
37
Maxwell’s wave equations
A.
38
Perturbation Theory
A.
38
.
1
Basic perturbation theory
A.
38
.
2
Ionization energy of helium
A.
38
.
3
Degenerate perturbation theory
A.
38
.
4
The Zeeman effect
A.
38
.
5
The Stark effect
A.
39
The relativistic hydrogen atom
A.
39
.
1
Introduction
A.
39
.
2
Fine structure
A.
39
.
3
Weak and intermediate Zeeman effect
A.
39
.
4
Lamb shift
A.
39
.
5
Hyperfine splitting
A.
40
Deuteron wave function
A.
41
Deuteron model
A.
41
.
1
The model
A.
41
.
2
The repulsive core
A.
41
.
3
Spin dependence
A.
41
.
4
Noncentral force
A.
41
.
5
Spin-orbit interaction
A.
42
Nuclear forces
A.
42
.
1
Basic Yukawa potential
A.
42
.
2
OPEP potential
A.
42
.
3
Explanation of the OPEP potential
A.
42
.
4
Multiple pion exchange and such
A.
43
Classical vibrating drop
A.
43
.
1
Basic definitions
A.
43
.
2
Kinetic energy
A.
43
.
3
Energy due to surface tension
A.
43
.
4
Energy due to Coulomb repulsion
A.
43
.
5
Frequency of vibration
A.
44
Relativistic neutrinos
A.
45
Fermi theory
A.
45
.
1
Form of the wave function
A.
45
.
2
Source of the decay
A.
45
.
3
Allowed or forbidden
A.
45
.
4
The nuclear operator
A.
45
.
5
Fermi’s golden rule
A.
45
.
6
Mopping up
A.
45
.
7
Electron capture
D. Derivations
D.
1
Generic vector identities
D.
2
Some Green’s functions
D.
2
.
1
The Poisson equation
D.
2
.
2
The screened Poisson equation
D.
3
Lagrangian mechanics
D.
3
.
1
Lagrangian equations of motion
D.
3
.
2
Hamiltonian dynamics
D.
3
.
3
Fields
D.
4
Lorentz transformation derivation
D.
5
Lorentz group property derivation
D.
6
Lorentz force derivation
D.
7
Derivation of the Euler formula
D.
8
Completeness of Fourier modes
D.
9
Momentum operators are Hermitian
D.
10
The curl is Hermitian
D.
11
Extension to three-dimensional solutions
D.
12
The harmonic oscillator solution
D.
13
The harmonic oscillator and uncertainty
D.
14
The spherical harmonics
D.
14
.
1
Derivation from the eigenvalue problem
D.
14
.
2
Parity
D.
14
.
3
Solutions of the Laplace equation
D.
14
.
4
Orthogonal integrals
D.
14
.
5
Another way to find the spherical harmonics
D.
14
.
6
Still another way to find them
D.
15
The hydrogen radial wave functions
D.
16
Constant spherical potentials derivations
D.
16
.
1
The eigenfunctions
D.
16
.
2
The Rayleigh formula
D.
17
Inner product for the expectation value
D.
18
Eigenfunctions of commuting operators
D.
19
The generalized uncertainty relationship
D.
20
Derivation of the commutator rules
D.
21
Solution of the hydrogen molecular ion
D.
22
Unique ground state wave function
D.
23
Solution of the hydrogen molecule
D.
24
Hydrogen molecule ground state and spin
D.
25
Number of boson states
D.
26
Density of states
D.
27
Radiation from a hole
D.
28
Kirchhoff’s law
D.
29
The thermionic emission equation
D.
30
Number of conduction band electrons
D.
31
Integral Schrödinger equation
D.
32
Integral conservation laws
D.
33
Quantum field derivations
D.
34
The adiabatic theorem
D.
35
The evolution of expectation values
D.
36
Photon wave function derivations
D.
36
.
1
Rewriting the energy integral
D.
36
.
2
Angular momentum states
D.
36
.
2
.
1
About the scalar modes
D.
36
.
2
.
2
Basic observations and eigenvalue problem
D.
36
.
2
.
3
Spherical form and net angular momentum
D.
36
.
2
.
4
Orthogonality and normalization
D.
36
.
2
.
5
Completeness
D.
36
.
2
.
6
Density of states
D.
36
.
2
.
7
Parity
D.
36
.
2
.
8
Orbital angular momentum of the states
D.
37
Forces by particle exchange derivations
D.
37
.
1
Classical energy minimization
D.
37
.
2
Quantum energy minimization
D.
37
.
3
Rewriting the Lagrangian
D.
37
.
4
Coulomb potential energy
D.
38
Time-dependent perturbation theory
D.
39
Selection rules
D.
40
Quantization of radiation derivations
D.
41
Derivation of the Einstein B coefficients
D.
42
Derivation of the Einstein A coefficients
D.
43
Multipole derivations
D.
43
.
1
Matrix element for linear momentum modes
D.
43
.
2
Matrix element for angular momentum modes
D.
43
.
3
Weisskopf and Moszkowski estimates
D.
44
Derivation of group velocity
D.
45
Motion through crystals
D.
45
.
1
Propagation speed
D.
45
.
2
Motion under an external force
D.
45
.
3
Free-electron gas with constant electric field
D.
46
Derivation of the WKB approximation
D.
47
Born differential cross section
D.
48
About Lagrangian multipliers
D.
49
The generalized variational principle
D.
50
Spin degeneracy
D.
51
Born-Oppenheimer nuclear motion
D.
52
Simplification of the Hartree-Fock energy
D.
53
Integral constraints
D.
54
Derivation of the Hartree-Fock equations
D.
55
Why the Fock operator is Hermitian
D.
56
Number of system eigenfunctions
D.
57
The particle energy distributions
D.
58
The canonical probability distribution
D.
59
Analysis of the ideal gas Carnot cycle
D.
60
Checks on the expression for entropy
D.
61
Chemical potential in the distributions
D.
62
Fermi-Dirac integrals at low temperature
D.
63
Angular momentum uncertainty
D.
64
Spherical harmonics by ladder operators
D.
65
How to make Clebsch-Gordan tables
D.
66
The triangle inequality
D.
67
Momentum of shells
D.
68
Awkward questions about spin
D.
69
More awkwardness about spin
D.
70
Emergence of spin from relativity
D.
71
Electromagnetic commutators
D.
72
Various electrostatic derivations.
D.
72
.
1
Existence of a potential
D.
72
.
2
The Laplace equation
D.
72
.
3
Egg-shaped dipole field lines
D.
72
.
4
Ideal charge dipole delta function
D.
72
.
5
Integrals of the current density
D.
72
.
6
Lorentz forces on a current distribution
D.
72
.
7
Field of a current dipole
D.
72
.
8
Biot-Savart law
D.
73
Orbital motion in a magnetic field
D.
74
Electron spin in a magnetic field
D.
75
Solving the NMR equations
D.
76
Harmonic oscillator revisited
D.
77
Impenetrable spherical shell
D.
78
Shell model quadrupole moment
D.
79
Derivation of perturbation theory
D.
80
Hydrogen ground state Stark effect
D.
81
Dirac fine structure Hamiltonian
D.
82
Classical spin-orbit derivation
D.
83
Expectation powers of
r
for hydrogen
D.
84
Band gap explanation derivations
N. Notes
N.
1
Why this book?
N.
2
History and wish list
N.
3
Nature and real eigenvalues
N.
4
Are Hermitian operators really like that?
N.
5
Why boundary conditions are tricky
N.
6
Is the variational approximation best?
N.
7
Shielding approximation limitations
N.
8
Why the s states have the least energy
N.
9
Explanation of the band gaps
N.
10
A less fishy story
N.
11
Better description of two-state systems
N.
12
Second quantization in other books
N.
13
Combining angular momentum factors
N.
14
The electric multipole problem
N.
15
A tenth of a googol in universes
N.
16
A single Slater determinant is not exact
N.
17
Generalized orbitals
N.
18
Correlation energy
N.
19
Ambiguities in electron affinity
N.
20
Why Floquet theory should be called so
N.
21
Superfluidity versus BEC
N.
22
The mechanism of ferromagnetism
N.
23
Fundamental assumption of statistics
N.
24
A problem if the energy is given
N.
25
The recipe of life
N.
26
Physics of the fundamental commutators
N.
27
Magnitude of components of vectors
N.
28
Adding angular momentum components
N.
29
Clebsch-Gordan tables are bidirectional
N.
30
Machine language Clebsch-Gordan tables
N.
31
Existence of magnetic monopoles
N.
32
More on Maxwell’s third law
N.
33
Setting the record straight on alignment
N.
34
NuDat 2 data selection
N.
35
Auger discovery
N.
36
Draft: Cage-of-Faraday proposal
Next:
1. Special Relativity [Draft]
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