List of equations in quantum mechanics

Template:Sidebar with collapsible lists This article summarizes equations in the theory of quantum mechanics.

Wavefunctions

A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is Template:Nowrap, also known as the reduced Planck constant or Dirac constant.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
Wavefunction	ψ, Ψ	To solve from the Schrödinger equation	varies with situation and number of particles
Wavefunction probability density	ρ	$\rho ={\left|\mathrm{\Psi }\right|}^2={\mathrm{\Psi }}^{\ast }\mathrm{\Psi }$	m−3	[L]−3
Wavefunction probability current	j	Non-relativistic, no external field: $\begin{array}{cc}𝐣& =\frac{-i\hslash }{2m}({\mathrm{\Psi }}^{\ast }\mathrm{\nabla }\mathrm{\Psi }-\mathrm{\Psi }\mathrm{\nabla }{\mathrm{\Psi }}^{\ast })\\ & =\frac{\hslash }{m}\mathrm{Im}\left({\mathrm{\Psi }}^{\ast }\mathrm{\nabla }\mathrm{\Psi }\right)=\mathrm{Re}\left({\mathrm{\Psi }}^{\ast }\frac{\hslash }{im}\mathrm{\nabla }\mathrm{\Psi }\right)\end{array}$ star * is complex conjugate	m−2⋅s−1	[T]−1 [L]−2

The general form of wavefunction for a system of particles, each with position r_i and z-component of spin s_{z i}. Sums are over the discrete variable s_z, integrals over continuous positions r.

For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary). Following are general mathematical results, used in calculations.

Property or effect	Nomenclature	Equation
Wavefunction for N particles in 3d	Template:Plainlist r = (r1, r2... rN) sz = (sz 1, sz 2, ..., sz N)	In function notation: $\mathrm{\Psi }=\mathrm{\Psi }(𝐫,{𝐬}_{𝐳},t)$ in bra–ket notation: $|\mathrm{\Psi }⟩=\sum _{s_{z1}}\sum _{s_{z2}}\cdots \sum _{s_{zN}}{\int }_{V_1}{\int }_{V_2}\cdots {\int }_{V_N}\mathrm{d}{𝐫}_1\mathrm{d}{𝐫}_2\cdots \mathrm{d}{𝐫}_N\mathrm{\Psi }|𝐫,{𝐬}_{𝐳}⟩$ for non-interacting particles: $\mathrm{\Psi }=\prod _{n=1}^N\mathrm{\Psi }({𝐫}_n,s_{zn},t)$
Position-momentum Fourier transform (1 particle in 3d)	Template:Plainlist Φ = momentum–space wavefunction Ψ = position–space wavefunction	$\begin{array}{cc}\mathrm{\Phi }(𝐩,s_z,t)& =\frac{1}{{\sqrt{2\pi \hslash }}^3}\underset{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}}{\int }{e}^{-i𝐩\cdot 𝐫/\hslash }\mathrm{\Psi }(𝐫,s_z,t){\mathrm{d}}^3𝐫\\ & \upharpoonleft \downharpoonright \\ \mathrm{\Psi }(𝐫,s_z,t)& =\frac{1}{{\sqrt{2\pi \hslash }}^3}\underset{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}}{\int }{e}^{+i𝐩\cdot 𝐫/\hslash }\mathrm{\Phi }(𝐩,s_z,t){\mathrm{d}}^3{𝐩}_n\end{array}$
General probability distribution	Template:Plainlist Vj = volume (3d region) particle may occupy, P = Probability that particle 1 has position r1 in volume V1 with spin sz1 and particle 2 has position r2 in volume V2 with spin sz2, etc.	$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}{\int }_{V_N}\cdots {\int }_{V_2}{\int }_{V_1}{\left|\mathrm{\Psi }\right|}^2{\mathrm{d}}^3{𝐫}_1{\mathrm{d}}^3{𝐫}_2\cdots {\mathrm{d}}^3{𝐫}_N$
General normalization condition		$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\underset{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}}{\int }\cdots \underset{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}}{\int }\underset{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}}{\int }{\left|\mathrm{\Psi }\right|}^2{\mathrm{d}}^3{𝐫}_1{\mathrm{d}}^3{𝐫}_2\cdots {\mathrm{d}}^3{𝐫}_N=1$

Equations

Wave–particle duality and time evolution

Property or effect	Nomenclature	Equation
Planck–Einstein equation and de Broglie wavelength relations	Template:Plainlist P = (E/c, p) is the four-momentum, K = (ω/c, k) is the four-wavevector, E = energy of particle ω = 2πf is the angular frequency and frequency of the particle ħ = h/2π are the Planck constants c = speed of light	$𝐏=(E/c,𝐩)=\hslash (\omega /c,𝐤)=\hslash 𝐊$
Schrödinger equation	Template:Plainlist Ψ = wavefunction of the system Ĥ = Hamiltonian operator, E = energy eigenvalue of system i is the imaginary unit t = time	General time-dependent case: $i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=\hat{H}\mathrm{\Psi }$ Time-independent case: $\hat{H}\mathrm{\Psi }=E\mathrm{\Psi }$
Heisenberg equation	Template:Plainlist  = operator of an observable property [ ] is the commutator $⟨⟩$ denotes the average	$\frac{d}{dt}\hat{A}(t)=\frac{i}{\hslash }[\hat{H},\hat{A}(t)]+\frac{\partial \hat{A}(t)}{\partial t}$
Time evolution in Heisenberg picture (Ehrenfest theorem)	Template:Plainlist m = mass, V = potential energy, r = position, p = momentum, Template:Endplainlist of a particle.	$\frac{d}{dt}⟨\hat{A}⟩=\frac{1}{i\hslash }⟨[\hat{A},\hat{H}]⟩+⟨\frac{\partial \hat{A}}{\partial t}⟩$ For momentum and position; $m\frac{d}{dt}⟨𝐫⟩=⟨𝐩⟩$ $\frac{d}{dt}⟨𝐩⟩=-⟨\mathrm{\nabla }V⟩$

Non-relativistic time-independent Schrödinger equation

Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.

	One particle	N particles
One dimension	$\hat{H}=\frac{{\hat{p}}^2}{2m}+V(x)=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)$	$\begin{array}{cc}\hat{H}& =\sum _{n=1}^N\frac{{\hat{p}}_n^2}{2m_n}+V(x_1,x_2,\cdots x_N)\\ & =-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}\frac{{\partial }^2}{\partial x_n^2}+V(x_1,x_2,\cdots x_N)\end{array}$ where the position of particle n is xn.
	$E\mathrm{\Psi }=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}\mathrm{\Psi }+V\mathrm{\Psi }$	$E\mathrm{\Psi }=-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}\frac{{\partial }^2}{\partial x_n^2}\mathrm{\Psi }+V\mathrm{\Psi }.$
	$\mathrm{\Psi }(x,t)=\psi (x)e^{-iEt/\hslash }.$ There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): $\Vert \psi {\Vert }^2=\int |\psi (x){|}^{2}dx.$	$\mathrm{\Psi }=e^{-iEt/\hslash }\psi (x_1,x_2\cdots x_N)$ for non-interacting particles $\mathrm{\Psi }=e^{-iEt/\hslash }\prod _{n=1}^N\psi (x_n),V(x_1,x_2,\cdots x_N)=\sum _{n=1}^{N}V(x_n).$
Three dimensions	$\begin{array}{cc}\hat{H}& =\frac{\widehat{𝐩}\cdot \widehat{𝐩}}{2m}+V(𝐫)\\ & =-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(𝐫)\end{array}$ where the position of the particle is r = (x, y, z).	$\begin{array}{cc}\hat{H}& =\sum _{n=1}^N\frac{{\widehat{𝐩}}_n\cdot {\widehat{𝐩}}_n}{2m_n}+V({𝐫}_1,{𝐫}_2,\cdots {𝐫}_N)\\ & =-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}{\mathrm{\nabla }}_n^2+V({𝐫}_1,{𝐫}_2,\cdots {𝐫}_N)\end{array}$ where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is ${\mathrm{\nabla }}_n^2=\frac{{\partial }^2}{{\partial x_n}^2}+\frac{{\partial }^2}{{\partial y_n}^2}+\frac{{\partial }^2}{{\partial z_n}^2}$
	$E\mathrm{\Psi }=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }+V\mathrm{\Psi }$	$E\mathrm{\Psi }=-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}{\mathrm{\nabla }}_n^2\mathrm{\Psi }+V\mathrm{\Psi }$
	$\mathrm{\Psi }=\psi (𝐫)e^{-iEt/\hslash }$	$\mathrm{\Psi }=e^{-iEt/\hslash }\psi ({𝐫}_1,{𝐫}_2\cdots {𝐫}_N)$ for non-interacting particles $\mathrm{\Psi }=e^{-iEt/\hslash }\prod _{n=1}^N\psi ({𝐫}_n),V({𝐫}_1,{𝐫}_2,\cdots {𝐫}_N)=\sum _{n=1}^{N}V({𝐫}_n)$

Non-relativistic time-dependent Schrödinger equation

Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.

	One particle	N particles
One dimension	$\hat{H}=\frac{{\hat{p}}^2}{2m}+V(x,t)=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V(x,t)$	$\begin{array}{cc}\hat{H}& =\sum _{n=1}^N\frac{{\hat{p}}_n^2}{2m_n}+V(x_1,x_2,\cdots x_N,t)\\ & =-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}\frac{{\partial }^2}{\partial x_n^2}+V(x_1,x_2,\cdots x_N,t)\end{array}$ where the position of particle n is xn.
	$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}\mathrm{\Psi }+V\mathrm{\Psi }$	$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}\frac{{\partial }^2}{\partial x_n^2}\mathrm{\Psi }+V\mathrm{\Psi }.$
	$\mathrm{\Psi }=\mathrm{\Psi }(x,t)$	$\mathrm{\Psi }=\mathrm{\Psi }(x_1,x_2\cdots x_N,t)$
Three dimensions	$\begin{array}{cc}\hat{H}& =\frac{\widehat{𝐩}\cdot \widehat{𝐩}}{2m}+V(𝐫,t)\\ & =-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(𝐫,t)\end{array}$	$\begin{array}{cc}\hat{H}& =\sum _{n=1}^N\frac{{\widehat{𝐩}}_n\cdot {\widehat{𝐩}}_n}{2m_n}+V({𝐫}_1,{𝐫}_2,\cdots {𝐫}_N,t)\\ & =-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}{\mathrm{\nabla }}_n^2+V({𝐫}_1,{𝐫}_2,\cdots {𝐫}_N,t)\end{array}$
	$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\mathrm{\Psi }+V\mathrm{\Psi }$	$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }=-\frac{{\hslash }^2}{2}\sum _{n=1}^N\frac{1}{m_n}{\mathrm{\nabla }}_n^2\mathrm{\Psi }+V\mathrm{\Psi }$ This last equation is in a very high dimension, so the solutions are not easy to visualize.
	$\mathrm{\Psi }=\mathrm{\Psi }(𝐫,t)$	$\mathrm{\Psi }=\mathrm{\Psi }({𝐫}_1,{𝐫}_2,\cdots {𝐫}_N,t)$

Photoemission

Property/Effect	Nomenclature	Equation
Photoelectric equation	Template:Plainlist Kmax = Maximum kinetic energy of ejected electron (J) h = Planck constant f = frequency of incident photons (Hz = s−1) φ, Φ = Work function of the material the photons are incident on (J)	$K_{\mathrm{m}\mathrm{a}\mathrm{x}}=hf-\mathrm{\Phi }$
Threshold frequency and Work function	Template:Plainlist φ, Φ = Work function of the material the photons are incident on (J) f0, ν0 = Threshold frequency (Hz = s−1)	Can only be found by experiment. The De Broglie relations give the relation between them: $\varphi =h{f}_0$
Photon momentum	Template:Plainlist p = momentum of photon (kg m s−1) f = frequency of photon (Hz = s−1) λ = wavelength of photon (m)	The De Broglie relations give: $p=hf/c=h/\lambda$

Quantum uncertainty

Property or effect	Nomenclature	Equation
Heisenberg's uncertainty principles	Template:Plainlist n = number of photons φ = wave phase [, ] = commutator	Position–momentum $\sigma (x)\sigma (p)\ge \frac{\hslash }{2}$ Energy-time $\sigma (E)\sigma (t)\ge \frac{\hslash }{2}$ Number-phase $\sigma (n)\sigma (\varphi )\ge \frac{\hslash }{2}$
Dispersion of observable	A = observables (eigenvalues of operator)	$\begin{array}{cc}\sigma (A{)}^2& =⟨(A-⟨A⟩{)}^2⟩\\ & =⟨A^2⟩-⟨A{⟩}^2\end{array}$
General uncertainty relation	A, B = observables (eigenvalues of operator)	$\sigma (A)\sigma (B)\ge \frac{1}{2}⟨i[\hat{A},\hat{B}]⟩$

Probability Distributions

Property or effect	Equation
Density of states	$N(E)=8\sqrt{2}\pi m^{3/2}{E}^{1/2}/h^3$
Fermi–Dirac distribution (fermions)	$P(E_i)=\frac{g(E_i)}{e^{(E-\mu )/kT}+1}$ where Template:Plainlist P(Ei) = probability of energy Ei g(Ei) = degeneracy of energy Ei (no of states with same energy) μ = chemical potential
Bose–Einstein distribution (bosons)	$P(E_i)=\frac{g(E_i)}{e^{(E_i-\mu )/kT}-1}$

Angular momentum

Property or effect	Nomenclature	Equation
Angular momentum quantum numbers	Template:Plainlist s = spin quantum number ms = spin magnetic quantum number ℓ = Azimuthal quantum number mℓ = azimuthal magnetic quantum number j = total angular momentum quantum number mj = total angular momentum magnetic quantum number	Spin: $\begin{array}{cc}& \Vert 𝐬\Vert =\sqrt{s(s+1)}\hslash \\ & m_s\in \{-s,-s+1\cdots s-1,s\}\end{array}$ Orbital: $\begin{array}{cc}& \ell \in \{0\cdots n-1\}\\ & m_{\ell }\in \{-\ell ,-\ell +1\cdots \ell -1,\ell \}\end{array}$ Total: $\begin{array}{cc}& j=\ell +s\\ & m_j\in \{|\ell -s|,|\ell -s|+1\cdots |\ell +s|-1,|\ell +s|\}\end{array}$
Angular momentum magnitudes	angular momementa:Template:Plainlist S = Spin, L = orbital, J = total	Spin magnitude: $|𝐒|=\hslash \sqrt{s(s+1)}$ Orbital magnitude: $|𝐋|=\hslash \sqrt{\ell (\ell +1)}$ Total magnitude: $𝐉=𝐋+𝐒$ $|𝐉|=\hslash \sqrt{j(j+1)}$
Angular momentum components		Spin: $S_z=m_s\hslash$ Orbital: $L_z=m_{\ell }\hslash$

Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

Property or effect	Nomenclature	Equation
orbital magnetic dipole moment	Template:Plainlist e = electron charge me = electron rest mass L = electron orbital angular momentum gTemplate:Ell = orbital Landé g-factor μB = Bohr magneton	${𝝁}_{\ell }=-e𝐋/2m_e=g_{\ell }\frac{{\mu }_B}{\hslash }𝐋$ z-component: ${\mu }_{\ell ,z}=-m_{\ell }{\mu }_B$
spin magnetic dipole moment	Template:Plainlist S = electron spin angular momentum gs = spin Landé g-factor	${𝝁}_s=-e𝐒/m_e=g_s\frac{{\mu }_B}{\hslash }𝐒$ z-component: ${\mu }_{s,z}=-e{S}_z/m_e=g_{s}e{S}_z/2m_e$
dipole moment potential	U = potential energy of dipole in field	$U=-𝝁\cdot 𝐁=-{\mu }_{z}B$

Hydrogen atom

Property or effect	Nomenclature	Equation
Energy level	Template:Plainlist En = energy eigenvalue n = principal quantum number e = electron charge me = electron rest mass ε0 = permittivity of free space h = Planck constant	$E_n=-m{e}^4/8{\epsilon }_0^2{h}^2{n}^2=-13.61\mathrm{e}\mathrm{V}/n^2$
Spectrum	λ = wavelength of emitted photon, during electronic transition from Ei to Ej	$\frac{1}{\lambda }=R(\frac{1}{n_j^2}-\frac{1}{n_i^2}),n_j<n_i$

See also

• Defining equation (physical chemistry)
• List of electromagnetism equations
• List of equations in classical mechanics
• List of equations in fluid mechanics
• List of equations in gravitation
• List of equations in nuclear and particle physics
• List of equations in wave theory
• List of photonics equations
• List of relativistic equations

Sources

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Further reading

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