Quantum Mechanics for Engineers
© Leon van Dommelen
Next:
A. Addenda
IV
. Supplementary Information
Subsections
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.
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:
A. Addenda
FAMU-FSU College of Engineering
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