Core Concepts Index

This page is a fast routing index for common concepts. Each entry gives a compact meaning and points toward the canonical home when one exists. It is not a substitute for the teaching pages.

Core Formalism

1. Quantum state — the mathematical object encoding preparation-dependent probabilities; see Quantum States.
2. State vector — a vector representative of a pure state; see State Vectors.
3. Ray — the physical equivalence class of nonzero state vectors differing by nonzero complex scale; see Rays and Global Phase.
4. Global phase — an overall phase with no observable effect; see Rays and Global Phase.
5. Relative phase — phase difference between components that can affect interference; see Superposition and Relative Phase.
6. Superposition — a linear combination of state vectors, physically meaningful through amplitudes and interference.
7. Wavefunction — coordinate-space representation of a state; see Coordinate Representation.
8. Hilbert space — complete inner-product space used for states; see Hilbert Spaces.
9. Inner product — structure giving amplitudes, norms, and orthogonality; see Inner Products.
10. Observable — quantity represented by a self-adjoint operator; see Observables.
11. Operator — linear map acting on states or wavefunctions.
12. Eigenvalue — possible sharp value associated with an eigenstate; see Eigenvalues and Eigenstates.
13. Eigenstate — state unchanged up to scale by an operator; see Eigenvalues and Eigenstates.
14. Projector — operator projecting onto a subspace; see Projectors.
15. Spectral decomposition — representation of an observable in terms of projectors and eigenvalues; see Spectral Decomposition.
16. Probability amplitude — complex number whose modulus squared gives probability in a specified context; see Probability Amplitudes.
17. Born rule — rule assigning probabilities from amplitudes; see Born Rule.
18. Expectation value — probability-weighted mean of measurement outcomes; see Expectation Values.
19. Variance — mean squared deviation of an observable from its expectation value.
20. Commutator — operator difference $AB-BA$ measuring noncommutativity; see Commutators.
21. Compatibility — simultaneous sharp measurability associated with commuting observables under appropriate hypotheses.
22. Canonical commutation relation — $[\hat x,\hat p]=i\hbar$ and its variants; see Canonical Commutation Relations.
23. Uncertainty relation — lower bound on simultaneous spreads of noncommuting observables; see General Uncertainty Relations.
24. Projective measurement — ideal measurement associated with orthogonal projectors; see Projective Measurement.
25. State update — conditional state change after measurement; see State Update Rule.
26. Hamiltonian — generator of time evolution and energy observable; see Hamiltonians.
27. Schrödinger equation — equation of motion for states; see Schrödinger Equation.
28. Unitary evolution — norm-preserving time evolution generated by self-adjoint Hamiltonians; see Unitary Time Evolution.
29. Tensor product — state-space construction for composite systems; see Tensor Products.
30. Entanglement — non-factorization of composite-system states; see Entangled States.
31. Density operator — positive trace-one operator representing pure or mixed states; see Density Operators.
32. Mixed state — state not representable by one ray alone; see Pure vs Mixed States.
33. Pure state — extremal quantum state represented by a ray or rank-one density operator.
34. Minimal postulates — compact formal rules for states, observables, dynamics, and measurement; see Minimal Postulates.

Mathematical Tools

35. Vector space — set closed under addition and scalar multiplication; see Vector Spaces and Dual Spaces.
36. Dual space — space of linear functionals, underlying bras.
37. Hermitian operator — finite-dimensional operator equal to its adjoint; see Hermitian Operators.
38. Unitary operator — operator preserving inner products; see Unitary Operators.
39. Pauli matrices — standard basis for spin-half and two-level Hermitian matrices; see Pauli Matrices.
40. Fourier transform — transform connecting position and momentum representations; see Fourier Transform.
41. Delta function — distribution representing idealized point support and continuum orthogonality; see Delta Function.
42. Wave packet — localized superposition of waves; see Wave Packets.
43. Sturm–Liouville problem — eigenvalue problem structure behind many bound systems; see Sturm–Liouville Theory.
44. Boundary condition — condition selecting allowed functions and spectra; see Boundary Conditions.
45. Spherical harmonic — angular eigenfunction on the sphere; see Spherical Harmonics.
46. SU(2) — Lie group underlying spin and angular momentum representations; see SU(2).
47. Angular momentum algebra — commutation relations and ladder structure of angular momentum; see Angular Momentum Algebra.
48. Matrix diagonalization — numerical or exact eigenvalue extraction; see Matrix Diagonalization.

Wave Mechanics And Canonical Systems

49. Coordinate representation — representation of states by position amplitudes; see Coordinate Representation.
50. Probability density — density integrated over regions to obtain probabilities; see Wavefunctions and Probability Density.
51. Time-dependent Schrödinger equation — coordinate-space equation of motion; see Time-Dependent Schrödinger Equation.
52. Time-independent Schrödinger equation — energy eigenvalue problem; see Time-Independent Schrödinger Equation.
53. Normalization convention — rule for scaling bound, continuum, box, or numerical states; see Normalization Conventions.
54. Free particle — particle with Hamiltonian $\hat p^2/(2m)$; see Free Particle.
55. Plane wave — momentum eigenfunction, not square-normalizable on the full line.
56. Gaussian wave packet — localized minimum-uncertainty free-particle packet; see Gaussian Wave Packets.
57. Infinite square well — hard-wall model of boundary quantization; see Infinite Square Well.
58. Finite square well — finite-confinement model with evanescent tails; see Finite Square Well.
59. Potential step — simplest scattering discontinuity; see Potential Step.
60. Tunneling — nonzero transmission through a finite classically forbidden region; see Rectangular Barrier Tunneling.
61. Harmonic oscillator — quadratic-potential model with evenly spaced levels; see Quantum Harmonic Oscillator.
62. Zero-point energy — nonzero ground-state energy of a quantum oscillator; see Zero-Point Energy.
63. Ladder operator — operator that raises or lowers oscillator or angular momentum quantum numbers; see Ladder-Operator Solution.
64. Hermite function — harmonic-oscillator eigenfunction built from a Hermite polynomial and Gaussian envelope.

Spin, Symmetry, And Information

65. Spin — intrinsic angular momentum degree of freedom.
66. Spin-half system — two-dimensional spin representation modeled by Pauli matrices.
67. Bloch sphere — geometric representation of pure two-level states; see Bloch Sphere: Wave-Mechanics Perspective.
68. Qubit — two-level quantum system used as a quantum information unit.
69. Measurement basis — orthonormal basis or projective decomposition used for a measurement.
70. POVM — generalized measurement described by positive operators summing to identity.
71. Quantum channel — completely positive trace-preserving map between states.
72. Partial trace — operation reducing a composite density operator to a subsystem.
73. Entropy — information measure such as von Neumann entropy.
74. Mutual information — measure of total correlations between systems.
75. Decoherence — suppression of interference by entanglement with uncontrolled degrees of freedom.

Approximation, Scattering, And Many-Body Concepts

76. Perturbation theory — expansion around a solvable Hamiltonian.
77. Degeneracy — multiple independent states with the same eigenvalue.
78. Selection rule — symmetry-imposed condition for matrix elements or transitions.
79. WKB approximation — semiclassical approximation for slowly varying potentials.
80. Scattering amplitude — complex amplitude encoding scattering into outgoing channels.
81. Cross section — effective scattering area or probability density in angle.
82. Resonance — enhanced response associated with quasi-bound or unstable structure.
83. Identical particles — particles requiring symmetric or antisymmetric state structure.
84. Boson — identical particle type with symmetric exchange structure.
85. Fermion — identical particle type with antisymmetric exchange structure.
86. Fock space — direct-sum space allowing variable particle number.
87. Second quantization — operator language for many-body and field modes.
88. Creation operator — operator adding a particle, quasiparticle, or excitation in the appropriate formalism.
89. Annihilation operator — operator removing a particle, quasiparticle, or excitation in the appropriate formalism.
90. Correlation function — expectation value probing relationships between observables at different points or times.

Atoms, Matter, Open Systems, And Bridges

91. Hydrogen atom — central Coulomb problem with bound orbitals and degeneracy.
92. Rigid rotor — angular kinetic-energy model for rotations.
93. Landau level — quantized cyclotron energy level in a magnetic field; see Landau Levels.
94. Berry phase — geometric phase acquired under adiabatic parameter transport.
95. Aharonov–Bohm effect — phase effect from electromagnetic potentials in field-free regions.
96. Lindblad equation — Markovian master equation for open quantum systems.
97. Path integral — formulation summing amplitudes over histories.
98. Propagator — kernel or operator carrying states between times or points.
99. Renormalization — scale-dependent reorganization of parameters and observables.
100. Classical limit — regime or approximation in which quantum predictions recover classical behavior.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.