Appendix I: Index

References are to chapter and section numbers (e.g., 5.3 means Chapter 5, Section 3). Bold entries indicate the primary treatment; italic entries indicate a Python toolkit module. Entries followed by (ex) refer to exercises; entries followed by (cs) refer to case studies.

A

Absorption, photon, 21.2, 21.4 Accidental degeneracy, 5.5, 9.4, 18.1 Action (classical), 31.1, 31.2 Adiabatic approximation, 32.1, 32.2 Adiabatic theorem, 32.1; proof, 32.1; conditions for validity, 32.3 Addition of angular momentum, 14.1--14.4; two spin-1/2, 14.2; spin-orbit, 14.3; general algorithm, 14.4 Adjoint operator, 6.4, 8.3 Aharonov-Bohm effect, 29.4, 32.4, 36.2 Alpha decay (tunneling), 3.5 (cs), 20.3 Angular momentum: orbital, 5.3, 12.1--12.4; spin, 13.1--13.5; addition, 14.1--14.4; algebra, 12.2; commutation relations, 12.2; eigenvalues, 12.3; ladder operators, 12.3; matrix representations, 12.4 Annihilation operator, 4.4, 8.6, 27.3, 34.2 Antimatter, prediction of, 34.1 Antisymmetric wave function, 15.2, 15.3, 15.4 Aspect experiments, 24.7, Appendix F Atomic units, Appendix C

B

Baker-Campbell-Hausdorff (BCH) lemma, 10.3 (ex), 7.5 Band structure, 26.2--26.3; gaps, 26.3; metals vs. insulators, 26.4 Band theory of solids, 26.1--26.6 Basis: orthonormal, 8.3; change of, 8.5; completeness, 8.4; continuous, 8.4; position, 8.4; momentum, 8.4 Beam splitter, 1.6, 7.3, 24.6, 27.5 Bell inequality, 24.4--24.6; CHSH form, 24.5; derivation, 24.4; experimental tests, 24.7; loopholes, 24.8 Bell states, 11.5, 24.3, 25.4 Bell's theorem, 24.4, Appendix F Berry connection, 32.2 Berry curvature, 32.2, 36.2 Berry phase, 32.2--32.3; examples, 32.3; relation to Aharonov-Bohm, 32.4 Blackbody radiation, 1.2, 1 (code); Planck formula, 1.2; Rayleigh-Jeans law, 1.2; ultraviolet catastrophe, 1.2; Wien's law, 1.2 Bloch sphere, 13.3, 25.1; rotations on, 13.4; visualization, spin.py Bloch's theorem, 26.1, 26.2 Bohr magneton, 13.5, 18.4, Appendix C Bohr model, 1.5, 5.5; limitations, 1.5, 5.5 Bohr radius, 1.5, 5.5, Appendix C Bohm interpretation, see Pilot-wave theory Born approximation, 22.3, 22.5 Born interpretation, 2.2, 2.3 Bose-Einstein condensation (BEC), 15.6, 26.6, 34.4, Appendix F Bose-Einstein statistics, 15.5, 15.6 Boson, 15.2, 15.5; examples, 15.2 Bound states, 3.2, 3.3; vs. scattering states, 3.4 Boundary conditions, 3.2, 3.3; matching, 3.3, 3.5 Bra, 8.2; anti-linearity, 8.2 Brillouin zone, 26.2, band_structure.py

C

Canonical commutation relation, 6.3; derivation, 6.3; significance, 6.3 Casimir effect, 34.3 Center of mass coordinates, 5.2 Central potential, 5.2; separation of variables, 5.2 Chern number, 36.2, topological.py CHSH inequality, see Bell inequality Classical limit, see Correspondence principle Classical turning point, 4.3, 20.1 Clebsch-Gordan coefficients, 14.2--14.4; tables, Appendix D; clebsch_gordan.py Coherent states, 4.6, 27.3; properties, 27.3; time evolution, 27.3; uncertainty, 4.6 Collapse, see Wave function collapse Commutator, 6.3; canonical, 6.3; angular momentum, 12.2; properties, 6.3 (ex) Compatible observables, 6.6, 9.3 Complementarity, 1.6, 28.2 Complete set of commuting observables (CSCO), 9.3, 12.4 Completeness relation, 8.4; discrete, 8.4; continuous, 8.4 Compton scattering, 1.4, Appendix F Compton wavelength, 1.4, Appendix C Condensed matter quantum mechanics, 26.1--26.6 Conservation laws, 10.1--10.5; and symmetry, 10.1 Constants, physical, Appendix C Continuity equation, 2.6 Continuous spectrum, 9.5 Copenhagen interpretation, 1.10, 28.1--28.2 Correspondence principle, 1.9, 4.5, 7.4; harmonic oscillator, 4.5; large quantum numbers, 4.5 Coulomb potential, 5.4, 5.5 Creation operator, 4.4, 8.6, 27.3, 34.2 Cross section: differential, 22.2; total, 22.2; Rutherford, 22.3

D

Darwin term, 18.2 Davisson-Germer experiment, 1.8, Appendix F De Broglie wavelength, 1.8, 1 (code) Decoherence, 33.2--33.4; timescales, 33.3; pointer basis, 33.3; and measurement, 33.4 Degeneracy, 3.4, 5.5; accidental, 5.5, 9.4; essential, 5.5; lifting of, 18.1 Degenerate perturbation theory, 18.1--18.4; algorithm, 18.1; fine structure, 18.2 Delta function potential, 3.4 Density matrix, 23.1--23.5; properties, 23.1; pure vs. mixed, 23.2; time evolution, 23.3; density_matrix.py Density of states, 21.3, 26.3 Dirac delta function, 8.4, 3.4 Dirac equation, 34.1; gamma matrices, 34.1; solutions, 34.1; hydrogen atom, 34.1 Dirac notation, 8.1--8.6; bra, 8.2; ket, 8.1; bracket, 8.2; outer product, 8.3 Double-slit experiment, 1.6, 7.3, 28.1 Dynamical phase, 32.1

E

Edge states, 36.3, topological.py Ehrenfest's theorem, 2.7, 7.4; derivation, 7.4 Eigenfunction, see Eigenstate Eigenvalue, 6.2, 9.1; continuous, 9.5; degenerate, 9.4; non-degenerate, 9.1 Eigenstate, 6.2, 9.1; completeness, 9.2; orthogonality, 9.1 Einstein A and B coefficients, 21.4 Electromagnetic transitions, 21.4; selection rules, 21.4 Electron configuration, 16.2, periodic_table.py Electron spin, see Spin Emission, photon: spontaneous, 21.4; stimulated, 21.4 Energy eigenstate, see Stationary state Energy levels: hydrogen atom, 5.5; harmonic oscillator, 4.3; infinite well, 3.2; finite well, 3.3 Entanglement, 11.4, 24.1--24.8; Bell states, 11.5; entropy, 23.4; measures, 24.2; entanglement.py EPR paradox, 24.2--24.3; Bohm version, 24.2; original argument, 24.2; Appendix F Evanescent wave, 3.5 Exchange degeneracy, 15.1 Exchange interaction, 15.3, 16.3 Exchange operator, 15.1 Exchange symmetry, 15.1--15.2 Exclusion principle, see Pauli exclusion principle Expectation value, 2.4, 8.3; time dependence, 7.4

F

Fermi energy, 15.5, 26.4 Fermi's golden rule, 21.3; derivation, 21.3; applications, 21.4; transitions.py Fermi-Dirac statistics, 15.5 Fermion, 15.2, 15.5; examples, 15.2 Feynman diagrams, 37.3 (preview) Feynman path integral, see Path integral Fine structure, 18.2; constant ($\alpha$), 18.2, Appendix C; hydrogen, 18.2; spin-orbit, 18.2; relativistic correction, 18.2; Darwin term, 18.2 Finite square well, 3.3; even/odd solutions, 3.3; transcendental equation, 3.3 Fock space, 34.2 Fock state, see Number state Fourier transform, 2.4, 7.3, 9.5 Franck-Hertz experiment, 1.5, Appendix F Free particle, 2.5, 7.2; propagator, 31.2 Functional (path integral), 31.1

G

Galilean invariance, 10.5 (ex) Gauge transformation, 29.4, 32.4 Gaussian wave packet, 7.2; spreading, 7.2; group velocity, 7.2 Generalized uncertainty principle, 6.7 Geometric phase, see Berry phase Good quantum numbers, 14.3, 18.2 Gram-Schmidt orthogonalization, 8.3, 9.4 Green's function, 22.3 Ground state: harmonic oscillator, 4.3; hydrogen, 5.5; infinite well, 3.2 Group velocity, 7.2

H

Hamiltonian, 2.3, 6.1; harmonic oscillator, 4.2; hydrogen, 5.2; matrix form, 8.5 Harmonic oscillator, see Quantum harmonic oscillator Hartree-Fock method, 16.4, 31.5 (preview) Heisenberg picture, 7.5; equations of motion, 7.5 Heisenberg uncertainty principle, see Uncertainty principle Helium atom, 15.3, 16.2, 19.2 Hermite polynomials, 4.3, Appendix B; generating function, 4.3 (ex); recursion, 4.3; Rodrigues formula, 4.3 Hermitian operator, 6.4; eigenvalues real, 6.4; eigenstates orthogonal, 6.4 Hidden variables, see Local hidden variable theory Higgs boson, 32 (cs), 37.3, Appendix F Hilbert space, 2.3, 8.1; finite-dimensional, 8.5; infinite-dimensional, 8.4; tensor product, 11.1 Hong-Ou-Mandel effect, 27.5, quantum_optics.py Hund's rules, 16.3 Hydrogen atom, 5.1--5.7; Bohr model, 1.5; degeneracy, 5.5; energy levels, 5.5; fine structure, 18.2; ground state, 5.5; orbitals, 5.6; quantum numbers, 5.5; radial equation, 5.4; selection rules, 21.4; spectrum, 5.7; hydrogen.py; capstone simulation, 38 Hyperfine structure, 18.3

I

Identical particles, 15.1--15.6 Incompatible observables, 6.6, 6.7 Infinite square well, 3.2; energy levels, 3.2; wave functions, 3.2; time evolution, 7.2 (ex) Inner product, 2.4, 8.2; properties, 8.2; Schwarz inequality, 8.2 (ex) Interference, quantum, 1.6, 7.3; single-photon, 1.6; two-slit, 1.6 Interpretation of quantum mechanics, 28.1--28.5; Copenhagen, 28.1; many-worlds, 28.4; pilot-wave, 28.3; consistent histories, 28.5; relational, 28.5

K

Ket, 8.1 Klein-Gordon equation, 34.1 (preview) Klein paradox, 34.1 Kramers degeneracy, 10.4 Kronig-Penney model, 26.2, band_structure.py

L

Ladder operators: harmonic oscillator, 4.4, 8.6; angular momentum, 12.3 Laguerre polynomials (associated), 5.4, Appendix B Lamb shift, 18.3 Lande $g$-factor, 18.4 Laplacian: Cartesian, 5.1; spherical, 5.1 Legendre polynomials (associated), 5.3, Appendix B Lindblad equation, 33.3, open_systems.py Linear operator, 6.1 Local hidden variable theory, 24.3; refuted, 24.7 Local realism, 24.3, 24.7 Loophole-free Bell tests, 24.8, Appendix F

M

Mach-Zehnder interferometer, 7.3, 27.5 Magnetic moment, 13.5, 18.4 Many-worlds interpretation, 28.4 Matrix mechanics, 8.5 Matrix representation of operators, 8.5, 12.4 Measurement: postulate, 6.5; problem, 28.1--28.5; repeated, 6.5; sequential Stern-Gerlach, 6.5, 13.4; measurement.py Mixed state, 23.2 Momentum: operator, 2.4, 6.1; eigenstate, 8.4; representation, 8.4

N

No-cloning theorem, 25.3 No-go theorems: Bell, 24.4; no-cloning, 25.3; Kochen-Specker, 28.2 Node (wave function), 3.2, 5.6 Normalization, 2.3; box, 9.5; delta function, 8.4 Number operator, 4.4; eigenvalues, 4.4 Number state, see Fock state Numerov method, 3 (code), solvers/numerov.py

O

Observable, 6.1; compatible, 6.6; incompatible, 6.6 Open quantum systems, 33.1--33.5 Operator: adjoint, 6.4; commutator, 6.3; Hermitian, 6.4; linear, 6.1; projection, 8.3; unitary, 7.1 Orbital angular momentum, see Angular momentum Orbitals, atomic, 5.6, 16.2; s, p, d, f, 5.6 Orthogonality, 3.2, 8.3, 9.1 Orthonormal basis, 8.3 Outer product, 8.3

P

Parity operator, 6 (ex), 10.2; eigenvalues, 10.2; selection rules, 21.4 Partial trace, 23.4; and entanglement, 23.4 Partial wave analysis, 22.4--22.5; phase shifts, 22.4; scattering.py Path integral, 31.1--31.4; free particle, 31.2; harmonic oscillator, 31.3; path_integral.py Pauli exclusion principle, 15.3, 16.1 Pauli matrices, 13.2; algebra, 13.2; properties, 13.2 Periodic potential, 26.1--26.2 Periodic table, 16.1--16.4; Aufbau principle, 16.2; electron configurations, 16.2; periodic_table.py Perturbation theory: non-degenerate, 17.1--17.4; degenerate, 18.1--18.4; time-dependent, 21.1--21.4; first-order correction, 17.2; second-order correction, 17.3; perturbation.py Phase shift (scattering), 22.4 Phase velocity, 7.2 Phonon, 4.6, 26.5, 34.3 Photoelectric effect, 1.3, Appendix F Photon, 1.3; energy, 1.3; momentum, 1.4; spin, 27.1 Pilot-wave theory, 28.3 Planck's constant, 1.2, Appendix C Planck distribution, 1.2 Potential: central, 5.2; Coulomb, 5.4; delta function, 3.4; finite well, 3.3; harmonic, 4.1; infinite well, 3.2; periodic, 26.1; step, 3.5 Probability amplitude, 1.6, 2.2 Probability current, 2.6 Probability density, 2.2 Projection operator, 8.3, 9.2 Propagator, 7.3, 31.2

Q

Quantization: energy, 1.2, 3.2; angular momentum, 5.3, 12.3; spin, 13.1 Quantum chromodynamics (QCD), 37.3 (preview) Quantum circuit, 25.2, qubit.py; gates, 25.2 Quantum computing, 25.1--25.6; algorithms, 25.5; error correction, 35; gates, 25.2; qubits, 25.1 Quantum electrodynamics (QED), 18.3, 37.2 Quantum entanglement, see Entanglement Quantum error correction, 35.1--35.5; bit-flip code, 35.2; Shor code, 35.3; Steane code, 35.4; error_correction.py Quantum field theory (QFT), 37.1--37.4; preview, 34.3, 37 Quantum gate, 25.2; Hadamard, 25.2; CNOT, 25.2; Toffoli, 25.2; T gate, 25.2 Quantum harmonic oscillator, 4.1--4.6; algebraic solution, 4.4; analytic solution, 4.3; coherent states, 4.6; energy levels, 4.3; ground state, 4.3; harmonic_oscillator.py Quantum information, 25.1--25.6 Quantum key distribution (QKD), 25.6, 36.4 Quantum number, 1.5, 5.5, 12.3, 13.1; good, 14.3, 18.2; principal, 5.5; orbital, 5.5; magnetic, 5.5; spin, 13.1 Quantum optics, 27.1--27.6; coherent states, 27.3; Fock states, 27.2; quantum_optics.py Quantum revival, 7.4 Quantum simulation toolkit (QST), see Progressive Project entries for each chapter; API reference, Appendix E Quantum supremacy, 25.6 (cs), 36.5, Appendix F Quantum teleportation, 25.4, qubit.py Quantum tunneling, see Tunneling Qubit, 25.1

R

Rabi oscillations, 21.2 Radial equation, 5.4; effective potential, 5.4; hydrogen, 5.4 Radial wave function, 5.4, 5.5; hydrogen, 5.5 Rayleigh-Jeans law, 1.2 Reduced mass, 5.2 Reflection coefficient, 3.5 Relativistic quantum mechanics, 34.1 Rotation operator, 10.2, 12.4 Rutherford scattering, 1.5, 22.3, Appendix F Rydberg constant, 1.5, 5.5

S

Scanning tunneling microscope (STM), 3.5 (cs) Scattering: amplitude, 22.2; Born approximation, 22.3; cross section, 22.2; matrix, 22.5; partial wave, 22.4; theory, 22.1--22.6; scattering.py Schrodinger equation: time-dependent, 2.1; time-independent, 2.5; 3D, 5.1; separability, 5.2 Schrodinger picture, 7.5 Schwarz inequality, 6.7, 8.2 (ex) Second quantization, 34.1--34.4; bosons, 34.2; fermions, 34.3; second_quantization.py Selection rules, 21.4; electric dipole, 21.4; magnetic dipole, 21.4 Separable state, 11.4 Separation of variables: 1D time, 2.5; 3D spherical, 5.2; angular, 5.3 Shor's algorithm, 25.5, 40 (capstone) Simultaneous eigenstates, 6.6, 9.3 Slater determinant, 15.4, identical_particles.py Solvay Conference (1927), 1.10, Appendix F Spectral decomposition, 9.2; continuous, 9.5 Spectral theorem, 9.2 Spherical coordinates, 5.1 Spherical harmonics, 5.3; table, 5.3; orthogonality, 5.3; Appendix B Spin, 13.1--13.5; spin-1/2, 13.1; spin-1, 13.5; addition, 14.1; precession, 13.4; spin.py Spin-orbit coupling, 18.2 Spin-statistics theorem, 15.2 Spontaneous emission, 21.4 Squeezed state, 27.4 Square well: finite, 3.3; infinite, 3.2 SSH model, 36.3, topological.py Stark effect, 18.3 Stationary state, 2.5, 7.1 Stern-Gerlach experiment, 1.7, 6.5, 13.1, 13.4, 28.1, Appendix F Stimulated emission, 21.4 Superposition, 1.6, 2.1; vs. mixed state, 23.2 Symmetry: and conservation, 10.1--10.5; continuous, 10.1; discrete, 10.2; transformation, 10.1

T

Tensor product, 11.1--11.5; definition, 11.1; composite systems, 11.2; tensor.py Time evolution, 7.1--7.5; operator, 7.1; Schrodinger vs. Heisenberg, 7.5; time_evolution.py Time-ordered exponential, 7.5, 21.1 Time reversal, 10.4 Topological insulator, 36.3 Topological phases, 36.1--36.4; Chern number, 36.2; edge states, 36.3; topological.py Transfer matrix, 3.5 Transition probability, 21.2 Transmission coefficient, 3.5 Trial wave function, 19.1 Tsirelson's bound, 24.5 Tunneling, 3.5; rate (WKB), 20.3; scanning tunneling microscope, 3.5 (cs)

U

Ultraviolet catastrophe, 1.2 Uncertainty principle, 6.7; energy-time, 7.4; position-momentum, 6.7 Unitary operator, 7.1 Unitary transformation, 10.1

V

Vacuum energy, 34.3, 37.2 Vacuum state, 4.4, 34.2 Variational method, 19.1--19.4; ground state, 19.1; excited states, 19.3; helium, 19.2; variational.py Vector potential, 29.4, 32.4 Von Neumann entropy, 23.3, density_matrix.py Von Neumann measurement scheme, 28.1

W

Wave function, 2.1; collapse, 6.5, 28.1; continuity, 2.3; normalization, 2.3; probability interpretation, 2.2; spreading, 7.2 Wave packet, 7.2; Gaussian, 7.2; spreading, 7.2; group velocity, 7.2; revival, 7.4 Wave-particle duality, 1.6 Wien's displacement law, 1.2 Wigner-Eckart theorem, 14.4 WKB approximation, 20.1--20.4; connection formulas, 20.2; tunneling, 20.3; quantization condition, 20.1; wkb.py Work function, 1.3

Z

Zeeman effect: normal, 18.4; anomalous, 18.4; Paschen-Back limit, 18.4 Zero-point energy, 4.3; Casimir effect, 34.3

Python Toolkit Modules

angular_momentum.py, Ch 12 band_structure.py, Ch 26 berry_phase.py, Ch 32 clebsch_gordan.py, Ch 14 constants.py, Ch 1 density_matrix.py, Ch 23 dirac_equation.py, Ch 34 (preview) entanglement.py, Ch 24 error_correction.py, Ch 35 fine_structure.py, Ch 18 frontiers.py, Ch 30 harmonic_oscillator.py, Ch 4 hilbert.py, Ch 8 hydrogen.py, Ch 5 identical_particles.py, Ch 15 measurement.py, Ch 28 open_systems.py, Ch 33 operators.py, Ch 6 path_integral.py, Ch 31 periodic_table.py, Ch 16 perturbation.py, Ch 17 potentials.py, Ch 3 qft_bridge.py, Ch 37 quantum_optics.py, Ch 27 qubit.py, Ch 25 scattering.py, Ch 22 second_quantization.py, Ch 34 solvers/numerov.py, Ch 3 spectral.py, Ch 9 spin.py, Ch 13 symmetry.py, Ch 10 tensor.py, Ch 11 time_evolution.py, Ch 7 topological.py, Ch 36 transitions.py, Ch 21 variational.py, Ch 19 wavefunction.py, Ch 2 wkb.py, Ch 20