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{{Quantum book backlink|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.

[[File:Collection of quantum formulas.jpg|thumb|450px]]
=== Key Formulas in Quantum Mechanics ===

This table lists key formulas in quantum mechanics, showing their names, expressions, and applications.
{| class="wikitable sortable"
|-
! Equation Name
! Formula
! Description
! Applications
|-
| [[Wikipedia:Angular momentum operator|Angular Momentum Components]]
| <math>L_z = m_{\ell} \hbar</math>
| Z-component of angular momentum.
| Quantized orbits.
|-
| [[Wikipedia:Compton scattering|Compton Effect: Change in Wavelength]]
| <math>\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)</math>
| Shift in photon wavelength after scattering.
| Compton scattering, evidence for photon momentum.
|-
| [[Wikipedia:Duane–Hunt law|Cutoff Wavelength]]
| <math>\lambda_{min} = \frac{h c}{K_0}</math>
| Minimum wavelength in bremsstrahlung.
| X-ray production.
|-
| [[Wikipedia:Matter wave|De Broglie Wavelength]]
| <math>p = \frac{h}{\lambda} = \hbar k</math>
| Wavelength associated with a particle's momentum.
| Matter waves, electron diffraction.
|-
| [[Wikipedia:Fermi–Dirac statistics|Occupancy Probability]]
| <math>P(E) = \frac{1}{e^{(E - E_F)/kT} + 1}</math>
| Fermi-Dirac distribution.
| Electron statistics in metals.
|-
| [[Wikipedia:Free electron model|Density of States]]
| <math>N(E) = 8 \sqrt{2} \pi m^{3/2} E^{1/2} / h^3</math>
| Number of states per energy interval (3D free electron gas).
| Solid-state physics, Fermi gas.
|-
| [[Wikipedia:Dirac equation|Dirac Equation]]
| <math>\left( \boldsymbol{\beta} m c^2 + c \sum_{k=1}^{3} \boldsymbol{\alpha}_k p_k \right) \Psi = i \hbar \frac{\partial}{\partial t} \Psi</math>
| Relativistic quantum equation for fermions.
| Particle physics, electrons.
|-
| [[Wikipedia:Electric dipole moment|Electric Dipole Potential Energy]]
| <math>V = -\mathbf{p} \cdot \mathbf{E}</math>
| Energy of dipole in electric field.
| Molecular physics.
|-
| [[Wikipedia:Coulomb potential|Electrostatic, Coulomb Potential Energy]]
| <math>V = \frac{q_1 q_2}{4 \pi \epsilon_0 r}</math>
| Coulomb potential.
| Atomic interactions.
|-
| [[Wikipedia:Free particle|Free Particle Schrödinger's Equation (1D)]]
| <math>-\frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d} x^2} \Psi = E \Psi</math>
| For free particle in 1D.
| Free particle motion.
|-
| [[Wikipedia:Free particle|Free Particle Schrödinger's Equation (3D)]]
| <math>-\frac{\hbar^2}{2m} \nabla^2 \Psi = E \Psi</math>
| For free particle in 3D.
| Scattering problems.
|-
| [[Wikipedia:Quantum harmonic oscillator|Harmonic Oscillator Potential Energy]]
| <math>V = \frac{1}{2} k x^2</math>
| Potential for harmonic oscillator.
| Vibrational modes, quantum optics.
|-
| [[Wikipedia:Uncertainty principle|Heisenberg's Uncertainty Principle]]
| <math>\Delta x \Delta p_x \geq \frac{\hbar}{2}</math> <br /> <math>\Delta E \Delta t \geq \frac{\hbar}{2}</math>
| Limits on simultaneous knowledge of position/momentum and energy/time.
| Fundamental limit in measurements, quantum tunneling.
|-
| [[Wikipedia:Hydrogen atom|Hydrogen Atom, Orbital Energy]]
| <math>E_n = -\frac{m e^4}{8 \epsilon_0^2 h^2 n^2} = -\frac{13.6 \, \mathrm{eV}}{n^2}</math>
| Energy levels of hydrogen atom.
| Atomic spectroscopy, Bohr model.
|-
| [[Wikipedia:Radial distribution function|Hydrogen Atom, Radial Probability Density]]
| <math>P(r) = \frac{4 r^2}{a^3} e^{-2r/a}</math>
| Probability density for electron position (ground state).
| Atomic orbitals.
|-
| [[Wikipedia:Hydrogen spectral series|Hydrogen Atom Spectrum, Rydberg Equation]]
| <math>\frac{1}{\lambda} = R_H \left( \frac{1}{n_2^2} - \frac{1}{n_1^2} \right)</math>
| Wavelengths of spectral lines.
| Hydrogen emission/absorption spectra.
|-
| [[Wikipedia:Particle in a box|Infinite Potential Well Energy Levels]]
| <math>E_n = \left( \frac{h n}{2L} \right)^2 \frac{1}{2m}</math>
| Energy levels for particle in a box.
| Quantum confinement, nanostructures.
|-
| [[Wikipedia:Klein–Gordon equation|Klein-Gordon Equation]]
| <math>\left( -\frac{1}{c^2} \frac{\partial^2}{\partial t^2} + \nabla^2 \right) \Psi = \left( \frac{m_0 c}{\hbar} \right)^2 \Psi</math>
| Relativistic equation for bosons.
| Scalar particles.
|-
| [[Wikipedia:Probability current|Law of Probability Conservation for Quantum Mechanics]]
| <math>\frac{\partial}{\partial t} \int_V |\Psi|^2 \, \mathrm{d} V + \int_S \mathbf{j} \cdot \mathrm{d} \mathbf{A} = 0</math>
| Conservation of probability.
| Quantum dynamics.
|-
| [[Wikipedia:Magnetic moment|Magnetic Dipole Potential Energy]]
| <math>V = -\mathbf{m} \cdot \mathbf{B}</math>
| Energy of dipole in magnetic field.
| Magnetic resonance.
|-
| [[Wikipedia:Moseley's law|Moseley's Law]]
| <math>f = \frac{c}{\lambda} = M_{K_{\alpha}} (Z-1)^2</math> <br /> <math>M_{K_{\alpha}} = 2.47 \times 10^{15}</math> Hz
| Frequency of K-α X-ray line.
| Atomic number determination, X-ray spectroscopy.
|-
| [[Wikipedia:Born rule|Normalization Integral]]
| <math>\int_{\mathbf{r} \in R} |\Psi|^2 \, \mathrm{d} V = 1</math>
| Normalizes the wavefunction.
| Probability calculations.
|-
| [[Wikipedia:Particle in a box|One-Dimensional Box Potential Energy]]
| <math>V = \begin{cases} 0 & x \in [a, b] \\ \infty & x \notin [a, b] \end{cases}</math>
| Potential for particle in a box.
| Quantum wells.
|-
| [[Wikipedia:Zeeman effect|Orbital Electron Magnetic Dipole Components]]
| <math>\mathbf{\mu}_{orb, z} = -m_{\ell} \mu_B</math>
| Z-component of orbital magnetic moment.
| Zeeman effect.
|-
| [[Wikipedia:Electron magnetic moment|Orbital Electron Magnetic Dipole Moment]]
| <math>\mathbf{\mu}_{orb} = -e \mathbf{L} / 2m</math>
| Magnetic moment due to orbital motion.
| Atomic magnetism.
|-
| [[Wikipedia:Zeeman effect|Orbital, Electron Magnetic Dipole Moment Potential]]
| <math>U = -\mathbf{\mu}_{orb} \cdot \mathbf{B}_{ext} = -\mu_{orb, z} B_{ext}</math>
| Potential in external field.
| Magnetic interactions.
|-
| [[Wikipedia:Electron magnetic moment|Spin, Electron Magnetic Dipole Moment]]
| <math>\mathbf{\mu_s} = - \frac{e}{m}\mathbf{S} = - g \frac{e}{2m} \mathbf{S}</math>
| Spin magnetic moment.
| Electron spin resonance.
|-
| [[Wikipedia:Photoelectric effect|Photoelectric Effect: Maximum Kinetic Energy]]
| <math>E_{\mathrm{k\;\!max}} = hf - \Phi</math>
| Maximum kinetic energy of photoelectrons.
| Photoelectric effect experiments, solar cells.
|-
| [[Wikipedia:Photon#Physical_properties|Photon Momentum]]
| <math>p = \frac{hf}{c} = \frac{h}{\lambda}</math>
| Momentum of a photon.
| Quantum optics, Compton scattering.
|-
| [[Wikipedia:Planck relation|Planck–Einstein Equation]]
| <math>E = hf = \frac{hc}{\lambda}</math>
| Relates energy of a photon to its frequency or wavelength.
| Wave-particle duality, photon energy calculations.
|-
| [[Wikipedia:Planck's law|Planck's Radiation Law (Frequency Form)]]
| <math>I(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{\frac{h\nu}{kT}} - 1}</math>
| Spectral radiance for blackbody in frequency.
| Blackbody radiation, stellar spectra.
|-
| [[Wikipedia:Planck's law|Planck's Radiation Law (Wavelength Form)]]
| <math>I(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda kT}} - 1}</math>
| Spectral radiance for blackbody in wavelength.
| Thermal radiation analysis.
|-
| [[Wikipedia:Probability current|Probability Current (Non-Relativistic)]]
| <math>\mathbf{j} = \frac{\hbar}{2 m i} (\Psi^* \nabla \Psi - \Psi \nabla \Psi^*)</math>
| Flow of probability.
| Current in quantum systems.
|-
| [[Wikipedia:Probability density function|Probability Density Function]]
| <math>\rho(\mathbf{r}, t) = |\Psi(\mathbf{r}, t)|^2</math>
| Probability density.
| Locating particles.
|-
| [[Wikipedia:Schrödinger equation|Schrödinger's Equation (General Form)]]
| <math>\hat{H} \Psi = E \Psi</math>
| Fundamental equation of quantum mechanics.
| Solving quantum systems.
|-
| [[Wikipedia:Spin (physics)|Spin Angular Momentum Magnitude]]
| <math>S = \hbar \sqrt{s(s+1)}</math>
| Magnitude of spin.
| Particle spin properties.
|-
| [[Wikipedia:Spin (physics)|Spin Projection Quantum Number]]
| <math>m_s \in \left\{ -\frac{1}{2}, +\frac{1}{2} \right\}</math>
| Spin along z-axis for electrons.
| Spintronics, NMR.
|-
| [[Wikipedia:Schrödinger equation|Time-Dependent Schrödinger's Equation (1D)]]
| <math>\left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V \right) \Psi = i \hbar \frac{\partial}{\partial t} \Psi</math>
| Time evolution in 1D.
| Dynamics of quantum systems.
|-
| [[Wikipedia:Schrödinger equation|Time-Dependent Schrödinger's Equation (3D)]]
| <math>\left( -\frac{\hbar^2}{2m} \nabla^2 + V \right) \Psi = i \hbar \frac{\partial}{\partial t} \Psi</math>
| Time evolution in 3D.
| Quantum simulations.
|-
| [[Wikipedia:Schrödinger equation|Time-Independent Schrödinger's Equation (1D)]]
| <math>\left( -\frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d} x^2} + V \right) \Psi = E \Psi</math>
| Stationary states in 1D.
| Bound states, potentials.
|-
| [[Wikipedia:Schrödinger equation|Time-Independent Schrödinger's Equation (3D)]]
| <math>\left( -\frac{\hbar^2}{2m} \nabla^2 + V \right) \Psi = E \Psi</math>
| Stationary states in 3D.
| Atomic and molecular physics.
|-
| [[Wikipedia:Particle in a box|Wavefunction of a Trapped Particle, One Dimensional Box]]
| <math>\Psi_n(x) = A \sin \left( \frac{n \pi x}{L} \right)</math>
| Wavefunction for particle in a box.
| Bound states, quantum wells.
|-
| [[Wikipedia:Work function|Work Function]]
| <math>\Phi = hf_0</math>
| Minimum energy to eject an electron.
| Photoelectric effect, surface physics.
|}

===2. Organized by topic===

Below are the same formulas grouped

==Quantum mechanics (QM)==
{{Quantum mechanics}}

=== Core Dynamical Equations ===

Time-Dependent Schrödinger Equation
<math>i\hbar ,\partial_t \Psi = \hat{H}\Psi</math>

Time-Independent Schrödinger Equation
<math>\hat{H}\psi = E\psi</math>

Time-Evolution Operator
<math>U(t) = e^{-i\hat{H}t/\hbar}</math>

=== Operators and Measurement Theory ===

Canonical Commutation Relation (Heisenberg)
<math>[x,p] = i\hbar</math>

Expectation Value
<math>\langle A\rangle = \langle \psi |A| \psi\rangle</math>

Born Rule (Measurement Probability)
<math>P(a) = |\langle a|\psi\rangle|^2</math>

== Harmonic Oscillator ==

Annihilation Operator
<math>a = \frac{1}{\sqrt{2\hbar m\omega}},(m\omega x + i p)</math>

Energy Levels
<math>E_n = \hbar\omega\left(n + \tfrac12\right)</math>

=== Perturbation Theory & Quantum Transitions ===

First-Order Energy Correction
<math>E_n^{(1)} = \langle n|V|n\rangle</math>

Fermi Golden Rule (Transition Rate)
<math>\Gamma = \frac{2\pi}{\hbar},|V_{fi}|^2,\rho(E)</math>

=== Continuity Equation & Probability Current ===

Probability Current
<math>j = \frac{\hbar}{2mi}(\psi^\nabla\psi - \psi\nabla\psi^)</math>
==Open quantum systems==

* Density Matrix (Statistical Mixture) <math>\rho = \sum_i p_i,|\psi_i\rangle\langle\psi_i|</math>

* Lindblad Master Equation (Markovian Open Systems) <math>\dot{\rho} = -\tfrac{i}{\hbar}[\hat{H},\rho] + \sum_k \mathcal{D}[L_k]\rho</math>

* von Neumann Entropy <math>S = -\mathrm{Tr}(\rho\log\rho)</math>

==Quantum information science (QIS)==
<math>I(A:B)=S(A)+S(B)-S(AB)</math>

<math>\Phi(\rho)=\sum_k A_k\rho A_k^\dagger</math> (quantum channels)

*Bell states <math>|\psi^\pm\rangle</math>, <math>|\phi^\pm\rangle</math>

*CNOT gate definition

*Qubit superposition <math>|\psi\rangle=\alpha|0\rangle+\beta|1\rangle</math>

==Quantum optics (QO)==

<math>a, a^\dagger</math> creation–annihilation operators

<math>H_{\text{int}}=-\mathbf{d}\cdot\mathbf{E}</math>

*Coherent state <math>| \alpha \rangle = e^{-|\alpha|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}} |n\rangle</math>

*Jaynes–Cummings Hamiltonian <math>H=\hbar\omega a^\dagger a+\tfrac12\hbar\omega_0\sigma_z+g(a^\dagger\sigma_-+a\sigma_+)</math>

==Quantum statistical mechanics==

*Partition function <math>Z=\mathrm{Tr}(e^{-\beta H})</math>

*Thermal state <math>\rho_\beta = e^{-\beta H}/Z</math>

*Response function <math>\chi(\omega)</math>

==Quantum field theory (QFT)==

*Canonical commutation <math>[ \phi(x),\pi(y) ] = i\hbar\delta(x-y)</math>

*Klein–Gordon equation <math>(\Box + m^2)\phi = 0</math>

*Dirac Lagrangian

*Relativistic dispersion <math>E^2=p^2 c^2+m^2c^4</math>

==3. Multi column version==

<div style="column-count:3">

* <math>i\hbar \partial_t \Psi = H\Psi</math>
* <math>H\psi = E\psi</math>
* <math>\Delta x\,\Delta p \ge \hbar/2</math>
* <math>[x,p]=i\hbar</math>
* <math>P(a)=|\langle a|\psi\rangle|^2</math>
* <math>\rho=\sum p_i|\psi_i\rangle\langle\psi_i|</math>
* <math>S=-\mathrm{Tr}(\rho\log\rho)</math>
* <math>E_n=\hbar\omega(n+1/2)</math>
* <math>a=(m\omega x+ip)/\sqrt{2\hbar m\omega}</math>
* <math>\Gamma=\frac{2\pi}{\hbar}|V_{fi}|^2\rho(E)</math>
* <small><math>I(A:B)=S(A)+S(B)-S(AB)</math></small>
* Bell states <math>|\psi^\pm\rangle</math>
* CNOT <math>=|0\rangle\langle0|\otimes I + |1\rangle\langle1|\otimes X</math>
* <math>\dot{\rho}=-\frac{i}{\hbar}[H,\rho]+\sum\mathcal{D}[L]\rho</math>
* <math>[ \phi(x),\pi(y) ] = i\hbar\delta(x-y)</math>
* <math>(\Box+m^2)\phi=0</math>
* <math>Z=\mathrm{Tr}(e^{-\beta H})</math>

</div>

==4. Wave Packet spreading example==

Free particle dispersion: <math>\sigma_x(t)=\sigma_x(0)\sqrt{1+\left(\frac{\hbar t}{2m\sigma_x(0)^2}\right)^2}</math>
→ Used in cold-atom clouds, ultrafast electron microscopy.

===Two-level Rabi oscillation===

Population oscillation: <math>P_e(t)=\sin^2(\Omega t/2)</math>
→ Atomic clocks, qubit control.

===Harmonic oscillator example===

Ground state energy: <math>E_0=\tfrac12\hbar\omega</math>
→ Zero-point fluctuations in quantum optics.<br><br>
{| class="wikitable sortable"
! Formula !! Description !! Applications
|-
| <math>i\hbar \frac{\partial}{\partial t}\Psi = \hat{H}\Psi</math> || Time-dependent Schrödinger equation || Dynamics, atoms, molecules
|-
| <math>\hat{H}\psi = E\psi</math> || Time-independent Schrödinger equation || Spectra, tunneling, bound states
|-
| <math>\Delta x\,\Delta p \ge \frac{\hbar}{2}</math> || Heisenberg uncertainty || Measurement limits, wave packets
|-
| <math>[x,p]=i\hbar</math> || Canonical commutator || Quantization, oscillators
|-
| <math>\langle A \rangle = \langle \psi |A| \psi\rangle</math> || Expectation value || Predictions, statistics
|-
| <math>P(a)=|\langle a|\psi\rangle|^2</math> || Born rule || Measurement probabilities
|-
| <math>\hat{U}(t)=e^{-iHt/\hbar}</math> || Time-evolution operator || Quantum gates, scattering
|-
| <math>\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|</math> || Density matrix || Decoherence, open systems
|-
| <math>S=-\mathrm{Tr}(\rho\log\rho)</math> || von Neumann entropy || Entanglement, thermodynamics
|-
| <math>\mathrm{Tr}(\rho A)</math> || Expectation via density matrix || Ensembles, thermal states
|-
| <math>\frac{d\rho}{dt}=-\frac{i}{\hbar}[H,\rho]+\sum_k\mathcal{D}[L_k]\rho</math> || Lindblad master eq. || Decoherence, dissipation
|-
| <math>\mathcal{D}[L]\rho=L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}</math> || Dissipator || Relaxation, noise
|-
| <math>Z=\mathrm{Tr}(e^{-\beta H})</math> || Partition function || Thermodynamics, blackbody
|-
| <math>\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int dp\, e^{ipx/\hbar}\phi(p)</math> || Fourier relation || Wavepackets, scattering
|-
| <math>j=\frac{\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*)</math> || Probability current || Continuity, tunneling
|-
| <math>\hat{a}=\frac{1}{\sqrt{2\hbar m\omega}}(m\omega x+i p)</math> || Annihilation operator || QHO, quantum optics
|-
| <math>E_n=\hbar\omega(n+\tfrac12)</math> || HO spectrum || Phonons, cavities
|-
| <math>\phi_n(x)=\dots</math> || HO eigenfunctions || Basis for perturbation theory
|-
| <math>\hat{H}_{\text{spin}}=-\gamma\mathbf{B}\cdot\mathbf{S}</math> || Spin Hamiltonian || NMR, ESR, qubits
|-
| <math>\chi(\omega)=\int_0^\infty dt\,e^{i\omega t}C(t)</math> || Response function || Conductivity, noise
|-
| <math>k=\sqrt{2mE}/\hbar</math> || Free-particle wavenumber || Beams, dispersion
|-
| <math>\psi(x)=\sum_n c_n\phi_n(x)</math> || Basis expansion || Computation, spectral theory
|-
| <math>H=H_0+\lambda V</math> || Perturbation theory split || Approximations, resonances
|-
| <math>E_n^{(1)}=\langle n|V|n\rangle</math> || 1st-order energy shift || Stark, Zeeman effects
|-
| <math>\Gamma = \frac{2\pi}{\hbar}| \langle f |V| i \rangle |^2 \rho(E_f)</math> || Fermi golden rule || Transition rates
|-
| <math>(a|b)=\mathrm{Tr}(a^\dagger b)</math> || Hilbert-Schmidt inner product || Superoperators, channels
|-
| <math>\Phi(\rho)=\sum_k A_k\rho A_k^\dagger</math> || CPTP map (quantum channel) || Noise, quantum info
|-
| <math>I(A:B)=S(A)+S(B)-S(AB)</math> || Mutual information || Correlations, QIT
|-
| <math>|\psi^\pm\rangle=\frac{1}{\sqrt2}(|01\rangle\pm|10\rangle)</math> || Bell states || Entanglement, teleportation
|-
| <math>U_{\text{CNOT}} = |0\rangle\langle0|\otimes I + |1\rangle\langle1|\otimes X</math> || CNOT gate || Quantum computing
|-
| <math>[ \phi(x),\pi(y)] = i\hbar\delta(x-y)</math> || Canonical QFT commutator || Field quantization
|-
| <math>E^2=p^2 c^2 + m^2 c^4</math> || Relativistic dispersion || QFT, particles
|-
| <math>\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi</math> || Dirac Lagrangian || Fermions, QED
|-
| <math>\Box \phi + m^2\phi = 0</math> || Klein-Gordon eq. || Bosons, relativistic waves
|}

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]

{{Sourceattribution|Quantum Formulas Collection|1}}

Physics:Quantum Formulas Collection - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template