Physics:Quantum Formulas Collection

←Physics:Quantum basics

Introduction

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

See also

Search

Core pathway

1. Physics:Quantum basics
2. Physics:Quantum mechanics
3. Physics:Quantum Mathematical Foundations of Quantum_Theory
4. Physics:Quantum Interpretations of quantum mechanics
5. Physics:Quantum Atomic structure and spectroscopy
6. Physics:Quantum Open quantum systems
7. Physics:Quantum Statistical mechanics
8. Physics:Quantum Kinetic theory
9. Physics:Plasma physics (fusion context)
10. Physics:Tokamak physics
11. Physics:Tokamak edge physics and recycling asymmetries

---

Full contents

Foundations

1. Physics:Quantum basics
2. Physics:Quantum mechanics
3. Physics:Quantum mechanics measurements
4. Physics:Quantum Mathematical Foundations of Quantum_Theory

Conceptual and interpretations

5. Physics:Quantum Interpretations of quantum mechanics
6. Physics:Quantum A Spooky Action at a Distance
7. Physics:Quantum A Walk Through the Universe
8. Physics:Quantum: The Secret of Cohesion: How Waves Hold Matter Together

Mathematical structure and systems

9. Physics:Quantum Exactly solvable quantum systems
10. Physics:Quantum Formulas Collection
11. Physics:Quantum A Matter Of Size
12. Physics:Quantum Symmetry in quantum mechanics
13. Physics:Quantum Matter Elements and Particles

Atomic and spectroscopy

14. Physics:Quantum Atomic structure and spectroscopy

Wavefunctions and modes

15. Physics:Number of independent spatial modes in a spherical volume

Quantum information and computing

16. Physics:Quantum information theory
17. Physics:Quantum Computing Algorithms in the NISQ Era
18. Physics:Quantum_Noisy_Qubits

Quantum optics and experiments

19. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
20. Physics:Quantum optics beam splitter experiments
21. Physics:Quantum Ultra fast lasers
22. Physics:Quantum Experimental quantum physics
23. Template Quantum optics operators

Open quantum systems

24. Physics:Quantum Open quantum systems

Quantum field theory

25. Physics:Quantum field theory (QFT) basics

Statistical mechanics and kinetic theory



26. Physics:Quantum Statistical mechanics
27. Physics:Quantum Kinetic theory

Plasma and fusion physics

28. Physics:Plasma physics (fusion context)
29. Physics:Tokamak physics
30. Physics:Tokamak edge physics and recycling asymmetries
• Hierarchy of modern physics models showing the progression from quantum statistical mechanics to kinetic theory and plasma physics, culminating in tokamak edge transport and recycling asymmetries.

Timeline

31. Physics:Quantum mechanics/Timeline
32. Physics:Quantum_mechanics/Timeline/Quiz/

Advanced and frontier topics

33. Physics:Quantum Supersymmetry
34. Physics:Quantum Black hole thermodynamics
35. Physics:Quantum Holographic principle
36. Physics:Quantum gravity
37. Physics:Quantum De Sitter invariant special relativity
38. Physics:Quantum Doubly special relativity

Author: Harold Foppele