Core Formulas Index

This is a quick formula checklist. Each line is intentionally compact: formula, assumption, and where to continue. Full derivations belong on canonical pages or formula cards.

States And Probability

1. State normalization: $\langle \psi|\psi\rangle=1$. Assumes a normalizable pure state; see State Vectors.
2. Wavefunction normalization: $\int |\psi(x)|^2\,dx=1$. Assumes square normalization; see Normalization Conventions.
3. Three-dimensional normalization: $\int |\psi(\mathbf r)|^2\,d^3r=1$. Use the correct coordinate measure.
4. Born rule, discrete: $P(a_n)=|\langle a_n|\psi\rangle|^2$. Assumes a normalized state and nondegenerate eigenstate notation; see Born Rule.
5. Born rule, projector form: $P(\Pi)=\langle\psi|\Pi|\psi\rangle$. Handles degenerate subspaces; see Projectors.
6. Born rule, density operator: $P(\Pi)=\operatorname{Tr}(\rho\Pi)$. Assumes $\rho$ is positive and trace one.
7. Position probability: $P(a\le x\le b)=\int_a^b|\psi(x)|^2\,dx$. Assumes one-dimensional position representation.
8. Probability density: $\rho(x,t)=|\psi(x,t)|^2$. Density must be integrated over a region.
9. Expectation value: $\langle A\rangle=\langle\psi|\hat A|\psi\rangle$. Pure-state form; see Expectation Values.
10. Density-operator expectation: $\langle A\rangle=\operatorname{Tr}(\rho A)$. Mixed-state and pure-state compatible.
11. Variance: $(\Delta A)^2=\langle A^2\rangle-\langle A\rangle^2$. Assumes $A$ is an observable with finite moments; see Variance and Standard Deviation.
12. Standard deviation: $\Delta A=\sqrt{\langle A^2\rangle-\langle A\rangle^2}$. Nonnegative by definition; see Variance and Standard Deviation.
13. Pure density operator: $\rho=|\psi\rangle\langle\psi|$. Assumes normalized $|\psi\rangle$.
14. Mixture: $\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|$. Requires $p_i\ge 0$ and $\sum_i p_i=1$.
15. Trace normalization: $\operatorname{Tr}\rho=1$. Required for density operators.
16. Purity: $\gamma=\operatorname{Tr}(\rho^2)$. Pure states have $\gamma=1$; mixed states have smaller purity in finite dimension.
17. Projector idempotence: $\Pi^2=\Pi$. Assumes an exact orthogonal projector.
18. Projector complement: $\Pi_\perp=I-\Pi$. Assumes $\Pi$ is a projector.
19. Completeness, discrete basis: $\sum_n |n\rangle\langle n|=I$. Assumes complete orthonormal basis.
20. Completeness, position basis: $\int dx\,|x\rangle\langle x|=I$. Distributional identity; see Coordinate Representation.

Operators And Dynamics

21. Commutator: $[A,B]=AB-BA$. Order matters; see Commutators.
22. Anticommutator: $\{A,B\}=AB+BA$. Common in spin and fermionic contexts.
23. Canonical commutation relation: $[\hat x,\hat p]=i\hbar$. Requires compatible domains; see Canonical Commutation Relations.
24. Robertson uncertainty: $\Delta A\,\Delta B\ge \frac12|\langle[A,B]\rangle|$. Assumes suitable domains; see General Uncertainty Relations.
25. Position-space momentum: $\hat p=-i\hbar\,d/dx$. Boundary conditions matter.
26. Hamiltonian from potential: $\hat H=\hat p^2/(2m)+V(\hat x)$. Nonrelativistic single particle.
27. Schrödinger equation: $i\hbar\,d|\psi\rangle/dt=\hat H|\psi\rangle$. Closed-system state evolution.
28. Coordinate TDSE: $i\hbar\partial_t\psi=-(\hbar^2/2m)\nabla^2\psi+V\psi$. Scalar potential, nonrelativistic particle.
29. Time-independent equation: $\hat H\psi_n=E_n\psi_n$. Time-independent Hamiltonian.
30. Stationary phase: $\psi_n(t)=\psi_n(0)e^{-iE_nt/\hbar}$. Energy eigenstate only.
31. Time-evolution operator: $|\psi(t)\rangle=U(t,t_0)|\psi(t_0)\rangle$. Closed-system unitary evolution.
32. Time-independent unitary: $U(t)=e^{-iHt/\hbar}$. Assumes time-independent $H$; see Time-Evolution Operator.
33. Unitarity: $U^\dagger U=I$. Preserves inner products.
34. Heisenberg equation: $dA_H/dt=(i/\hbar)[H,A_H]+\partial A_H/\partial t$. Heisenberg picture.
35. Spectral decomposition: $A=\sum_n a_n\Pi_n$. Discrete spectrum form; see Spectral Decomposition.
36. Operator function: $f(A)=\sum_n f(a_n)\Pi_n$. Assumes spectral decomposition.
37. Tensor product inner product: $\langle a\otimes b|c\otimes d\rangle=\langle a|c\rangle\langle b|d\rangle$. Product vectors.
38. Partial trace definition: $\rho_A=\operatorname{Tr}_B\rho_{AB}$. Subsystem state.
39. Commutator product rule: $[A,BC]=[A,B]C+B[A,C]$. Useful algebra identity.
40. Baker-Campbell-Hausdorff first terms: $e^A B e^{-A}=B+[A,B]+\frac12[A,[A,B]]+\cdots$. Requires convergence or formal use.

Wave Mechanics

41. Coordinate wavefunction: $\psi(x)=\langle x|\psi\rangle$. Position representation.
42. Position inner product: $\langle\phi|\psi\rangle=\int \phi^*(x)\psi(x)\,dx$. One-dimensional form.
43. Momentum wavefunction: $\phi(p)=\langle p|\psi\rangle$. Momentum representation.
44. Fourier transform convention: $\phi(p)=(2\pi\hbar)^{-1/2}\int e^{-ipx/\hbar}\psi(x)\,dx$. Site convention; see Fourier Transform Conventions.
45. Inverse Fourier transform: $\psi(x)=(2\pi\hbar)^{-1/2}\int e^{ipx/\hbar}\phi(p)\,dp$. Same convention.
46. Delta normalization: $\langle p|p'\rangle=\delta(p-p')$. Continuum normalization.
47. Probability current: $j=(\hbar/2mi)(\psi^*\psi'-\psi\psi'^*)$. One-dimensional scalar-potential form.
48. Continuity equation: $\partial_t\rho+\nabla\cdot\mathbf j=0$. Closed conservative dynamics.
49. Plane wave energy: $E=\hbar^2k^2/(2m)$. Free nonrelativistic particle.
50. Plane wave current: $j=(\hbar k/m)|A|^2$. For $Ae^{ikx}$.
51. Free-particle dispersion: $\omega(k)=\hbar k^2/(2m)$. Nonrelativistic.
52. Group velocity: $v_g=d\omega/dk=\hbar k/m=p/m$. Free particle.
53. Phase velocity: $v_{\text{ph}}=\omega/k=\hbar k/(2m)$. Not the particle velocity.
54. Gaussian uncertainty: $\Delta x\,\Delta p=\hbar/2$. For the standard minimum-uncertainty Gaussian; see Position-Momentum Uncertainty.
55. Gaussian spreading: $\sigma_x(t)=\sigma_x(0)\sqrt{1+(\hbar t/(2m\sigma_x^2(0)))^2}$. Free Gaussian packet.

Canonical One-Dimensional Systems

56. Infinite well eigenfunctions: $\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)$. Hard walls on $0<x<L$.
57. Infinite well energies: $E_n=n^2\pi^2\hbar^2/(2mL^2)$. $n=1,2,\ldots$.
58. Infinite well momenta squared: $\langle p^2\rangle_n=n^2\pi^2\hbar^2/L^2$. Stationary eigenstate.
59. Finite well inside wave number: $q=\sqrt{2m(E+V_0)}/\hbar$. For $V=-V_0$ inside.
60. Finite well decay constant: $\kappa=\sqrt{-2mE}/\hbar$. Bound states with outside zero.
61. Finite well even equation: $q\tan(qa)=\kappa$. Symmetric attractive well.
62. Finite well odd equation: $-q\cot(qa)=\kappa$. Symmetric attractive well.
63. Potential step reflection: $R=((k-q)/(k+q))^2$. Step with $E>V_0$.
64. Potential step transmission: $T=4kq/(k+q)^2$. Current ratio for $E>V_0$.
65. Barrier decay constant: $\kappa=\sqrt{2m(V_0-E)}/\hbar$. Tunneling regime.
66. Rectangular barrier transmission: $T=[1+V_0^2\sinh^2(\kappa a)/(4E(V_0-E))]^{-1}$. Equal potentials on both sides.
67. Opaque barrier estimate: $T\approx 16E(V_0-E)e^{-2\kappa a}/V_0^2$. Assumes $\kappa a\gg 1$.

Harmonic Oscillator

68. Oscillator Hamiltonian: $H=p^2/(2m)+m\omega^2x^2/2$. One-dimensional oscillator.
69. Oscillator length: $\ell=\sqrt{\hbar/(m\omega)}$. Natural length scale.
70. Oscillator energies: $E_n=\hbar\omega(n+1/2)$. $n=0,1,2,\ldots$.
71. Oscillator ground state: $\psi_0(x)=(\pi\ell^2)^{-1/4}e^{-x^2/(2\ell^2)}$. Normalized.
72. Oscillator eigenfunctions: $\psi_n=(2^n n!)^{-1/2}(\pi\ell^2)^{-1/4}H_n(x/\ell)e^{-x^2/(2\ell^2)}$. Standard Hermite convention.
73. Annihilation operator: $a=2^{-1/2}(x/\ell+i\ell p/\hbar)$. Operator hats often suppressed in quick notation.
74. Creation operator: $a^\dagger=2^{-1/2}(x/\ell-i\ell p/\hbar)$. Adjoint of $a$.
75. Oscillator commutator: $[a,a^\dagger]=1$. Canonical oscillator algebra.
76. Number operator: $N=a^\dagger a$. Eigenvalues $n=0,1,2,\ldots$.
77. Oscillator Hamiltonian algebraic form: $H=\hbar\omega(N+1/2)$. Ladder-operator solution.
78. Lowering action: $a|n\rangle=\sqrt n\,|n-1\rangle$. Number states.
79. Raising action: $a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle$. Number states.
80. Ground-state uncertainty: $\Delta x=\ell/\sqrt2$ and $\Delta p=\sqrt{m\hbar\omega/2}$. Harmonic oscillator ground state.

Spin And Angular Momentum

81. Pauli product: $\sigma_i\sigma_j=\delta_{ij}I+i\epsilon_{ijk}\sigma_k$. Pauli matrices.
82. Pauli commutator: $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$. Spin-half algebra.
83. Pauli anticommutator: $\{\sigma_i,\sigma_j\}=2\delta_{ij}I$. Pauli matrices.
84. Spin-half operator: $S_i=(\hbar/2)\sigma_i$. Spin-half convention.
85. Angular momentum commutator: $[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$. Angular momentum algebra.
86. Angular momentum ladder: $J_\pm=J_x\pm iJ_y$. Standard definition.
87. Ladder action: $J_\pm|j,m\rangle=\hbar\sqrt{j(j+1)-m(m\pm1)}\,|j,m\pm1\rangle$. Standard angular momentum basis.
88. Angular momentum eigenvalues: $J^2|j,m\rangle=\hbar^2j(j+1)|j,m\rangle$. Standard basis.
89. Magnetic quantum number: $J_z|j,m\rangle=\hbar m|j,m\rangle$. $m=-j,\ldots,j$.
90. Spherical harmonic orthonormality: $\int Y_{\ell}^{m*}Y_{\ell'}^{m'}\,d\Omega=\delta_{\ell\ell'}\delta_{mm'}$. Standard angular measure.
91. Orbital angular eigenvalue: $L^2Y_\ell^m=\hbar^2\ell(\ell+1)Y_\ell^m$. Spherical harmonics.
92. Azimuthal eigenvalue: $L_zY_\ell^m=\hbar mY_\ell^m$. Standard convention.

Density, Information, And Open-System Starters

93. von Neumann entropy: $S(\rho)=-\operatorname{Tr}(\rho\log\rho)$. Log base must be specified.
94. Reduced state: $\rho_A=\operatorname{Tr}_B(\rho_{AB})$. Composite systems.
95. Product state density operator: $\rho_{AB}=\rho_A\otimes\rho_B$. No correlations.
96. Maximally mixed qubit: $\rho=I/2$. Two-dimensional system.
97. Bloch representation: $\rho=(I+\mathbf r\cdot\boldsymbol\sigma)/2$. Qubit state, $|\mathbf r|\le 1$.
98. Unitary channel: $\rho\mapsto U\rho U^\dagger$. Closed-system state update.
99. Kraus map: $\rho\mapsto\sum_k M_k\rho M_k^\dagger$. Trace preserving if $\sum_k M_k^\dagger M_k=I$.
100. Lindblad form: $\dot\rho=-i[H,\rho]/\hbar+\sum_k(L_k\rho L_k^\dagger-\{L_k^\dagger L_k,\rho\}/2)$. Markovian master-equation form.

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.