Physics:Quantum Formulas Collection

← Back to Mathematical structure and systems

This page provides a list of the most important formulas in quantum mechanics, useful as a quick reference for students, teachers, and researchers. The formulas are organized by topic and include names, mathematical expressions, and short explanations of what they mean and how they are used. While this collection focuses on key results, science is always evolving, and new discoveries may override or extend these formulas. You, the reader, are welcome to suggest additions or corrections to keep this resource up to date.

Key Formulas in Quantum Mechanics

This table lists key formulas in quantum mechanics, showing their names, expressions, and applications.

Equation Name	Formula	Description	Applications
Angular Momentum Components	$L_z=m_{\ell }\hslash$	Z-component of angular momentum.	Quantized orbits.
Compton Effect: Change in Wavelength	$\mathrm{\Delta }\lambda =\frac{h}{m_{e}c}(1-\cos \theta )$	Shift in photon wavelength after scattering.	Compton scattering, evidence for photon momentum.
Cutoff Wavelength	${\lambda }_{min}=\frac{hc}{K_0}$	Minimum wavelength in bremsstrahlung.	X-ray production.
De Broglie Wavelength	$p=\frac{h}{\lambda }=\hslash k$	Wavelength associated with a particle's momentum.	Matter waves, electron diffraction.
Occupancy Probability	$P(E)=\frac{1}{e^{(E-E_F)/kT}+1}$	Fermi-Dirac distribution.	Electron statistics in metals.
Density of States	$N(E)=8\sqrt{2}\pi m^{3/2}{E}^{1/2}/h^3$	Number of states per energy interval (3D free electron gas).	Solid-state physics, Fermi gas.
Dirac Equation	$(\mathit{\beta }m{c}^2+c\sum _{k=1}^3{\mathit{\alpha }}_k{p}_k)\mathrm{\Psi }=i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }$	Relativistic quantum equation for fermions.	Particle physics, electrons.
Electric Dipole Potential Energy	$V=-\mathbf{𝐩}\cdot \mathbf{𝐄}$	Energy of dipole in electric field.	Molecular physics.
Electrostatic, Coulomb Potential Energy	$V=\frac{q_1{q}_2}{4\pi {ϵ}_{0}r}$	Coulomb potential.	Atomic interactions.
Free Particle Schrödinger's Equation (1D)	$-\frac{{\hslash }^2}{2m}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\mathrm{\Psi }=E\mathrm{\Psi }$	For free particle in 1D.	Free particle motion.
Free Particle Schrödinger's Equation (3D)	$-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }=E\mathrm{\Psi }$	For free particle in 3D.	Scattering problems.
Harmonic Oscillator Potential Energy	$V=\frac{1}{2}k{x}^2$	Potential for harmonic oscillator.	Vibrational modes, quantum optics.
Heisenberg's Uncertainty Principle	$\mathrm{\Delta }x\mathrm{\Delta }{p}_x\ge \frac{\hslash }{2}$ $\mathrm{\Delta }E\mathrm{\Delta }t\ge \frac{\hslash }{2}$	Limits on simultaneous knowledge of position/momentum and energy/time.	Fundamental limit in measurements, quantum tunneling.
Hydrogen Atom, Orbital Energy	$E_n=-\frac{m{e}^4}{8{ϵ}_0^2{h}^2{n}^2}=-\frac{13.6\mathrm{e}\mathrm{V}}{n^2}$	Energy levels of hydrogen atom.	Atomic spectroscopy, Bohr model.
Hydrogen Atom, Radial Probability Density	$P(r)=\frac{4r^2}{a^3}{e}^{-2r/a}$	Probability density for electron position (ground state).	Atomic orbitals.
Hydrogen Atom Spectrum, Rydberg Equation	$\frac{1}{\lambda }=R_H(\frac{1}{n_2^2}-\frac{1}{n_1^2})$	Wavelengths of spectral lines.	Hydrogen emission/absorption spectra.
Infinite Potential Well Energy Levels	$E_n={\left(\frac{hn}{2L}\right)}^2\frac{1}{2m}$	Energy levels for particle in a box.	Quantum confinement, nanostructures.
Klein-Gordon Equation	$(-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}+{\mathrm{\nabla }}^2)\mathrm{\Psi }={\left(\frac{m_{0}c}{\hslash }\right)}^2\mathrm{\Psi }$	Relativistic equation for bosons.	Scalar particles.
Law of Probability Conservation for Quantum Mechanics	$\frac{\partial }{\partial t}\underset{V}{\int }|\mathrm{\Psi }{|}^2\mathrm{d}V+\underset{S}{\int }\mathbf{𝐣}\cdot \mathrm{d}\mathbf{𝐀}=0$	Conservation of probability.	Quantum dynamics.
Magnetic Dipole Potential Energy	$V=-\mathbf{𝐦}\cdot \mathbf{𝐁}$	Energy of dipole in magnetic field.	Magnetic resonance.
Moseley's Law	$f=\frac{c}{\lambda }=M_{K_{\alpha }}(Z-1{)}^2$ $M_{K_{\alpha }}=2.47\times 10^{15}$ Hz	Frequency of K-α X-ray line.	Atomic number determination, X-ray spectroscopy.
Normalization Integral	$\underset{\mathbf{𝐫}\in R}{\int }|\mathrm{\Psi }{|}^2\mathrm{d}V=1$	Normalizes the wavefunction.	Probability calculations.
One-Dimensional Box Potential Energy	$V=\{\begin{array}{ll}0& x\in [a,b]\\ \mathrm{\infty }& x\notin [a,b]\end{array}$	Potential for particle in a box.	Quantum wells.
Orbital Electron Magnetic Dipole Components	${\mathit{\mu }}_{orb,z}=-m_{\ell }{\mu }_B$	Z-component of orbital magnetic moment.	Zeeman effect.
Orbital Electron Magnetic Dipole Moment	${\mathit{\mu }}_{orb}=-e\mathbf{𝐋}/2m$	Magnetic moment due to orbital motion.	Atomic magnetism.
Orbital, Electron Magnetic Dipole Moment Potential	$U=-{\mathit{\mu }}_{orb}\cdot {\mathbf{𝐁}}_{ext}=-{\mu }_{orb,z}{B}_{ext}$	Potential in external field.	Magnetic interactions.
Spin, Electron Magnetic Dipole Moment	${\mathit{\mu }}_{\mathbf{𝐬}}=-\frac{e}{m}\mathbf{𝐒}=-g\frac{e}{2m}\mathbf{𝐒}$	Spin magnetic moment.	Electron spin resonance.
Photoelectric Effect: Maximum Kinetic Energy	$E_{\mathrm{k}\mathrm{m}\mathrm{a}\mathrm{x}}=hf-\mathrm{\Phi }$	Maximum kinetic energy of photoelectrons.	Photoelectric effect experiments, solar cells.
Photon Momentum	$p=\frac{hf}{c}=\frac{h}{\lambda }$	Momentum of a photon.	Quantum optics, Compton scattering.
Planck–Einstein Equation	$E=hf=\frac{hc}{\lambda }$	Relates energy of a photon to its frequency or wavelength.	Wave-particle duality, photon energy calculations.
Planck's Radiation Law (Frequency Form)	$I(\nu ,T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{kT}}-1}$	Spectral radiance for blackbody in frequency.	Blackbody radiation, stellar spectra.
Planck's Radiation Law (Wavelength Form)	$I(\lambda ,T)=\frac{2h{c}^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}$	Spectral radiance for blackbody in wavelength.	Thermal radiation analysis.
Probability Current (Non-Relativistic)	$\mathbf{𝐣}=\frac{\hslash }{2mi}({\mathrm{\Psi }}^{*}\mathrm{\nabla }\mathrm{\Psi }-\mathrm{\Psi }\mathrm{\nabla }{\mathrm{\Psi }}^{*})$	Flow of probability.	Current in quantum systems.
Probability Density Function	$\rho (\mathbf{𝐫},t)=|\mathrm{\Psi }(\mathbf{𝐫},t){|}^2$	Probability density.	Locating particles.
Schrödinger's Equation (General Form)	$\hat{H}\mathrm{\Psi }=E\mathrm{\Psi }$	Fundamental equation of quantum mechanics.	Solving quantum systems.
Spin Angular Momentum Magnitude	$S=\hslash \sqrt{s(s+1)}$	Magnitude of spin.	Particle spin properties.
Spin Projection Quantum Number	$m_s\in \{-\frac{1}{2},+\frac{1}{2}\}$	Spin along z-axis for electrons.	Spintronics, NMR.
Time-Dependent Schrödinger's Equation (1D)	$(-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V)\mathrm{\Psi }=i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }$	Time evolution in 1D.	Dynamics of quantum systems.
Time-Dependent Schrödinger's Equation (3D)	$(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V)\mathrm{\Psi }=i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }$	Time evolution in 3D.	Quantum simulations.
Time-Independent Schrödinger's Equation (1D)	$(-\frac{{\hslash }^2}{2m}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+V)\mathrm{\Psi }=E\mathrm{\Psi }$	Stationary states in 1D.	Bound states, potentials.
Time-Independent Schrödinger's Equation (3D)	$(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V)\mathrm{\Psi }=E\mathrm{\Psi }$	Stationary states in 3D.	Atomic and molecular physics.
Wavefunction of a Trapped Particle, One Dimensional Box	${\mathrm{\Psi }}_n(x)=A\sin \left(\frac{n\pi x}{L}\right)$	Wavefunction for particle in a box.	Bound states, quantum wells.
Work Function	$\mathrm{\Phi }=h{f}_0$	Minimum energy to eject an electron.	Photoelectric effect, surface physics.

2. Organized by topic

Below are the same formulas grouped

Quantum mechanics (QM)

Core Dynamical Equations

Time-Dependent Schrödinger Equation $i\hslash ,{\partial }_t\mathrm{\Psi }=\hat{H}\mathrm{\Psi }$

Time-Independent Schrödinger Equation $\hat{H}\psi =E\psi$

Time-Evolution Operator $U(t)=e^{-i\hat{H}t/\hslash }$

Operators and Measurement Theory

Canonical Commutation Relation (Heisenberg) $[x,p]=i\hslash$

Expectation Value $⟨A⟩=⟨\psi |A|\psi ⟩$

Born Rule (Measurement Probability) $P(a)=|⟨a|\psi ⟩{|}^2$

Harmonic Oscillator

Annihilation Operator $a=\frac{1}{\sqrt{2\hslash m\omega }},(m\omega x+ip)$

Energy Levels $E_n=\hslash \omega (n+\frac{1}{2})$

Perturbation Theory & Quantum Transitions

First-Order Energy Correction $E_n^{(1)}=⟨n|V|n⟩$

Fermi Golden Rule (Transition Rate) $\mathrm{\Gamma }=\frac{2\pi }{\hslash },|V_{fi}{|}^2,\rho (E)$

Continuity Equation & Probability Current

Probability Current $j=\frac{\hslash }{2mi}({\psi }^{\mathrm{\nabla }}\psi -\psi \mathrm{\nabla }{\psi }^{)}$

Open quantum systems

• Density Matrix (Statistical Mixture) $\rho =\sum _i{p}_i,|{\psi }_i⟩⟨{\psi }_i|$

• Lindblad Master Equation (Markovian Open Systems) $\dot{\rho }=-\frac{i}{\hslash }[\hat{H},\rho ]+\sum _k{𝒟}[L_k]\rho$

• von Neumann Entropy $S=-\mathrm{T}\mathrm{r}(\rho \log \rho )$

Quantum information science (QIS)

$I(A:B)=S(A)+S(B)-S(AB)$

$\mathrm{\Phi }(\rho )=\sum _k{A}_k\rho A_k^{†}$ (quantum channels)

• Bell states $|{\psi }^{\pm }⟩$, $|{\varphi }^{\pm }⟩$

• CNOT gate definition

• Qubit superposition $|\psi ⟩=\alpha |0⟩+\beta |1⟩$

Quantum optics (QO)

$a,a^{†}$ creation–annihilation operators

$H_{\text{int}}=-\mathbf{𝐝}\cdot \mathbf{𝐄}$

• Coherent state $|\alpha ⟩=e^{-|\alpha {|}^2/2}\sum _n\frac{{\alpha }^n}{\sqrt{n!}}|n⟩$

• Jaynes–Cummings Hamiltonian $H=\hslash \omega a^{†}a+\frac{1}{2}\hslash {\omega }_0{\sigma }_z+g(a^{†}{\sigma }_{-}+a{\sigma }_{+})$

Quantum statistical mechanics

• Partition function $Z=\mathrm{T}\mathrm{r}(e^{-\beta H})$

• Thermal state ${\rho }_{\beta }=e^{-\beta H}/Z$

• Response function $\chi (\omega )$

Quantum field theory (QFT)

• Canonical commutation $[\varphi (x),\pi (y)]=i\hslash \delta (x-y)$

• Klein–Gordon equation $(◻+m^2)\varphi =0$

• Dirac Lagrangian

• Relativistic dispersion $E^2=p^2{c}^2+m^2{c}^4$

3. Multi column version

• $i\hslash {\partial }_t\mathrm{\Psi }=H\mathrm{\Psi }$
• $H\psi =E\psi$
• $\mathrm{\Delta }x\mathrm{\Delta }p\ge \hslash /2$
• $[x,p]=i\hslash$
• $P(a)=|⟨a|\psi ⟩{|}^2$
• $\rho =\sum p_i|{\psi }_i⟩⟨{\psi }_i|$
• $S=-\mathrm{T}\mathrm{r}(\rho \log \rho )$
• $E_n=\hslash \omega (n+1/2)$
• $a=(m\omega x+ip)/\sqrt{2\hslash m\omega }$
• $\mathrm{\Gamma }=\frac{2\pi }{\hslash }|V_{fi}{|}^2\rho (E)$
• $I(A:B)=S(A)+S(B)-S(AB)$
• Bell states $|{\psi }^{\pm }⟩$
• CNOT $=|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes X$
• $\dot{\rho }=-\frac{i}{\hslash }[H,\rho ]+\sum {𝒟}[L]\rho$
• $[\varphi (x),\pi (y)]=i\hslash \delta (x-y)$
• $(◻+m^2)\varphi =0$
• $Z=\mathrm{T}\mathrm{r}(e^{-\beta H})$

4. Wave Packet spreading example

Free particle dispersion: ${\sigma }_x(t)={\sigma }_x(0)\sqrt{1+{\left(\frac{\hslash t}{2m{\sigma }_x(0{)}^2}\right)}^2}$ → Used in cold-atom clouds, ultrafast electron microscopy.

Two-level Rabi oscillation

Population oscillation: $P_e(t)={\sin }^2(\mathrm{\Omega }t/2)$ → Atomic clocks, qubit control.

Harmonic oscillator example

Ground state energy: $E_0=\frac{1}{2}\hslash \omega$ → Zero-point fluctuations in quantum optics.

Formula	Description	Applications
$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=\hat{H}\mathrm{\Psi }$	Time-dependent Schrödinger equation	Dynamics, atoms, molecules
$\hat{H}\psi =E\psi$	Time-independent Schrödinger equation	Spectra, tunneling, bound states
$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$	Heisenberg uncertainty	Measurement limits, wave packets
$[x,p]=i\hslash$	Canonical commutator	Quantization, oscillators
$⟨A⟩=⟨\psi |A|\psi ⟩$	Expectation value	Predictions, statistics
$P(a)=|⟨a|\psi ⟩{|}^2$	Born rule	Measurement probabilities
$\hat{U}(t)=e^{-iHt/\hslash }$	Time-evolution operator	Quantum gates, scattering
$\rho =\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|$	Density matrix	Decoherence, open systems
$S=-\mathrm{T}\mathrm{r}(\rho \log \rho )$	von Neumann entropy	Entanglement, thermodynamics
$\mathrm{T}\mathrm{r}(\rho A)$	Expectation via density matrix	Ensembles, thermal states
$\frac{d\rho }{dt}=-\frac{i}{\hslash }[H,\rho ]+\sum _k{𝒟}[L_k]\rho$	Lindblad master eq.	Decoherence, dissipation
${𝒟}[L]\rho =L\rho L^{†}-\frac{1}{2}\{L^{†}L,\rho \}$	Dissipator	Relaxation, noise
$Z=\mathrm{T}\mathrm{r}(e^{-\beta H})$	Partition function	Thermodynamics, blackbody
$\psi (x)=\frac{1}{\sqrt{2\pi \hslash }}\int dp{e}^{ipx/\hslash }\varphi (p)$	Fourier relation	Wavepackets, scattering
$j=\frac{\hslash }{2mi}({\psi }^{*}\mathrm{\nabla }\psi -\psi \mathrm{\nabla }{\psi }^{*})$	Probability current	Continuity, tunneling
$\hat{a}=\frac{1}{\sqrt{2\hslash m\omega }}(m\omega x+ip)$	Annihilation operator	QHO, quantum optics
$E_n=\hslash \omega (n+\frac{1}{2})$	HO spectrum	Phonons, cavities
${\varphi }_n(x)=\dots$	HO eigenfunctions	Basis for perturbation theory
${\hat{H}}_{\text{spin}}=-\gamma \mathbf{𝐁}\cdot \mathbf{𝐒}$	Spin Hamiltonian	NMR, ESR, qubits
$\chi (\omega )=\underset{0}{\overset{\mathrm{\infty }}{\int }}dt{e}^{i\omega t}C(t)$	Response function	Conductivity, noise
$k=\sqrt{2mE}/\hslash$	Free-particle wavenumber	Beams, dispersion
$\psi (x)=\sum _n{c}_n{\varphi }_n(x)$	Basis expansion	Computation, spectral theory
$H=H_0+\lambda V$	Perturbation theory split	Approximations, resonances
$E_n^{(1)}=⟨n|V|n⟩$	1st-order energy shift	Stark, Zeeman effects
$\mathrm{\Gamma }=\frac{2\pi }{\hslash }|⟨f|V|i⟩{|}^2\rho (E_f)$	Fermi golden rule	Transition rates
$(a|b)=\mathrm{T}\mathrm{r}(a^{†}b)$	Hilbert-Schmidt inner product	Superoperators, channels
$\mathrm{\Phi }(\rho )=\sum _k{A}_k\rho A_k^{†}$	CPTP map (quantum channel)	Noise, quantum info
$I(A:B)=S(A)+S(B)-S(AB)$	Mutual information	Correlations, QIT
$|{\psi }^{\pm }⟩=\frac{1}{\sqrt{2}}(|01⟩\pm |10⟩)$	Bell states	Entanglement, teleportation
$U_{\text{CNOT}}=|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes X$	CNOT gate	Quantum computing
$[\varphi (x),\pi (y)]=i\hslash \delta (x-y)$	Canonical QFT commutator	Field quantization
$E^2=p^2{c}^2+m^2{c}^4$	Relativistic dispersion	QFT, particles
${ℒ}=\overline{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi$	Dirac Lagrangian	Fermions, QED
$◻\varphi +m^2\varphi =0$	Klein-Gordon eq.	Bosons, relativistic waves

Table of contents (136 articles)

Index

Core theory

1. Foundations
2. Conceptual and interpretations
3. Mathematical structure and systems
4. Atomic and spectroscopy
5. Wavefunctions and modes
6. Quantum dynamics and evolution
7. Measurement and information
8. Quantum information and computing

Matter (by scale)

9. Matter (by scale)

Applications and extensions

10. Quantum optics and experiments
11. Open quantum systems
12. Quantum field theory
13. Statistical mechanics and kinetic theory
14. Condensed matter and solid-state physics
15. Plasma and fusion physics
16. Timeline
17. Advanced and frontier topics

Methods and tools

18. Methods and tools

Full contents

1. Foundations (7)↑

1. Physics:Quantum basics
2. Physics:Quantum Postulates
3. Physics:Quantum Hilbert space
4. Physics:Quantum Observables and operators
5. Physics:Quantum mechanics
6. Physics:Quantum mechanics measurements
7. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (10)↑

8. Physics:Quantum Interpretations of quantum mechanics
9. Physics:Quantum Wave–particle duality
10. Physics:Quantum Complementarity principle
11. Physics:Quantum Uncertainty principle
12. Physics:Quantum Measurement problem
13. Physics:Quantum Bell's theorem
14. Physics:Quantum Hidden variable theory
15. Physics:Quantum A Spooky Action at a Distance
16. Physics:Quantum A Walk Through the Universe
17. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

3. Mathematical structure and systems (11)↑

18. Physics:Quantum Density matrix
19. Physics:Quantum Exactly solvable quantum systems
20. Physics:Quantum Formulas Collection
21. Physics:Quantum A Matter Of Size
22. Physics:Quantum Symmetry in quantum mechanics
23. Physics:Quantum Angular momentum operator
24. Physics:Quantum Runge–Lenz vector
25. Physics:Quantum Approximation Methods
26. Physics:Quantum Matter Elements and Particles
27. Physics:Quantum Dirac equation
28. Physics:Quantum Klein–Gordon equation

4. Atomic and spectroscopy (11)↑

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

29. Physics:Quantum Atomic structure and spectroscopy
30. Physics:Quantum Hydrogen atom
31. Physics:Quantum Multi-electron atoms
32. Physics:Quantum Fine structure
33. Physics:Quantum Hyperfine structure
34. Physics:Quantum Isotopic shift
35. Physics:Quantum Zeeman effect
36. Physics:Quantum Stark effect
37. Physics:Quantum Spectral lines and series
38. Physics:Quantum Selection rules
39. Physics:Quantum Fermi's golden rule

5. Wavefunctions and modes (8)↑

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

40. Physics:Quantum Wavefunction
41. Physics:Quantum Superposition principle
42. Physics:Quantum Eigenstates and eigenvalues
43. Physics:Quantum Boundary conditions and quantization
44. Physics:Quantum Standing waves and modes
45. Physics:Quantum Normal modes and field quantization
46. Physics:Number of independent spatial modes in a spherical volume
47. Physics:Quantum Density of states

6. Quantum dynamics and evolution (9)↑

48. Physics:Quantum Time evolution
49. Physics:Quantum Schrödinger equation
50. Physics:Quantum Time-dependent Schrödinger equation
51. Physics:Quantum Stationary states
52. Physics:Quantum Perturbation theory
53. Physics:Quantum Time-dependent perturbation theory
54. Physics:Quantum Adiabatic theorem
55. Physics:Quantum Scattering theory
56. Physics:Quantum S-matrix

7. Measurement and information (6)↑

57. Physics:Quantum Measurement theory
58. Physics:Quantum Measurement operators
59. Physics:Quantum Projective measurement
60. Physics:Quantum POVM
61. Physics:Quantum Weak measurement
62. Physics:Quantum Measurement collapse

8. Quantum information and computing (6)↑

63. Physics:Quantum information theory
64. Physics:Quantum Qubit
65. Physics:Quantum Entanglement
66. Physics:Quantum Gates and circuits
67. Physics:Quantum Computing Algorithms in the NISQ Era
68. Physics:Quantum Noisy Qubits

9. Quantum optics and experiments (4)↑

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

69. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
70. Physics:Quantum optics beam splitter experiments
71. Physics:Quantum Ultra fast lasers
72. Physics:Quantum Experimental quantum physics
Template:Quantum optics operators

10. Open quantum systems (7)↑

73. Physics:Quantum Open systems
74. Physics:Quantum Master equation
75. Physics:Quantum Lindblad equation
76. Physics:Quantum Decoherence
77. Physics:Quantum Markovian dynamics
78. Physics:Quantum Non-Markovian dynamics
79. Physics:Quantum Trajectories

11. Quantum field theory (18)↑

Structural dependency map of quantum field theory.

80. Physics:Quantum field theory (QFT) basics
81. Physics:Quantum field theory (QFT) core
82. Physics:Quantum Fields and Particles
83. Physics:Quantum Second quantization
84. Physics:Quantum Harmonic Oscillator field modes
85. Physics:Quantum Creation and annihilation operators
86. Physics:Quantum vacuum fluctuations
87. Physics:Quantum Propagators in quantum field theory
88. Physics:Quantum Feynman diagrams
89. Physics:Quantum Path integral formulation
90. Physics:Quantum Renormalization in field theory
91. Physics:Quantum Renormalization group
92. Physics:Quantum Field Theory Gauge symmetry
93. Physics:Quantum Non-Abelian gauge theory
94. Physics:Quantum Electrodynamics (QED)
95. Physics:Quantum chromodynamics (QCD)
96. Physics:Quantum Electroweak theory
97. Physics:Quantum Standard Model

12. Statistical mechanics and kinetic theory (9)↑

98. Physics:Quantum Statistical mechanics
99. Physics:Quantum Partition function
100. Physics:Quantum Distribution functions
101. Physics:Quantum Liouville equation
102. Physics:Quantum Kinetic theory
103. Physics:Quantum Boltzmann equation
104. Physics:Quantum BBGKY hierarchy
105. Physics:Quantum Relaxation and thermalization
106. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (7)↑

107. Physics:Quantum Band structure
108. Physics:Quantum Fermi surfaces
109. Physics:Quantum Semiconductor physics
110. Physics:Quantum Phonons
111. Physics:Quantum Electron-phonon interaction
112. Physics:Quantum Superconductivity
113. Physics:Quantum Topological phases of matter

14. Plasma and fusion physics (8)↑

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

114. Physics:Quantum Fusion reactions and Lawson criterion
115. Physics:Quantum Plasma (fusion context)
116. Physics:Quantum Magnetic confinement fusion
117. Physics:Quantum Inertial confinement fusion
118. Physics:Quantum Plasma instabilities and turbulence
119. Physics:Quantum Tokamak core plasma
120. Physics:Quantum Tokamak edge physics and recycling asymmetries
121. Physics:Quantum Stellarator

15. Timeline (8)↑

122. Physics:Quantum mechanics/Timeline
123. Physics:Quantum mechanics/Timeline/Pre-quantum era
124. Physics:Quantum mechanics/Timeline/Old quantum theory
125. Physics:Quantum mechanics/Timeline/Modern quantum mechanics
126. Physics:Quantum mechanics/Timeline/Quantum field theory era
127. Physics:Quantum mechanics/Timeline/Quantum information era
128. Physics:Quantum mechanics/Timeline/Quantum technology era
129. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (6)↑

130. Physics:Quantum Supersymmetry
131. Physics:Quantum Black hole thermodynamics
132. Physics:Quantum Holographic principle
133. Physics:Quantum gravity
134. Physics:Quantum De Sitter invariant special relativity
135. Physics:Quantum Doubly special relativity

Author: Harold Foppele