D. Derivations

This appendix gives various derivations. Sometimes you need to see the derivation to judge whether a result is applicable in given circumstances. And some people like to see the derivation period.

Subsections

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.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. De­riva­tions

D. Derivations