List of equations in quantum mechanics

This article summarizes equations in the theory of quantum mechanics.

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	$\rho =\left\|\Psi \right\|^{2}=\Psi ^{*}\Psi$	m⁻³	[L]⁻³
Wavefunction probability current	j	Non-relativistic, no external field: ${\begin{aligned}\mathbf {j} &={\frac {-i\hbar }{2m}}\left(\Psi ^{}\nabla \Psi -\Psi \nabla \Psi ^{}\right)\\&={\frac {\hbar }{m}}\operatorname {Im} \left(\Psi ^{}\nabla \Psi \right)=\operatorname {Re} \left(\Psi ^{}{\frac {\hbar }{im}}\nabla \Psi \right)\end{aligned}}$ star * is complex conjugate	m⁻²⋅s⁻¹	[T]⁻¹ [L]⁻²

Property or effect	Nomenclature	Equation
Wavefunction for N particles in 3d	r = (r₁, r₂... r_N) s_z = (s_{z 1}, s_{z 2}, ..., s_{z N})	In function notation: $\Psi =\Psi \left(\mathbf {r} ,\mathbf {s_{z}} ,t\right)$ in bra–ket notation: $\|\Psi \rangle =\sum _{s_{z1}}\sum _{s_{z2}}\cdots \sum _{s_{zN}}\int _{V_{1}}\int _{V_{2}}\cdots \int _{V_{N}}\mathrm {d} \mathbf {r} _{1}\mathrm {d} \mathbf {r} _{2}\cdots \mathrm {d} \mathbf {r} _{N}\Psi \|\mathbf {r} ,\mathbf {s_{z}} \rangle$ for non-interacting particles: $\Psi =\prod _{n=1}^{N}\Psi \left(\mathbf {r} _{n},s_{zn},t\right)$
Position-momentum Fourier transform (1 particle in 3d)	Φ = momentum–space wavefunction Ψ = position–space wavefunction	${\begin{aligned}\Phi (\mathbf {p} ,s_{z},t)&={\frac {1}{{\sqrt {2\pi \hbar }}^{3}}}\int \limits _{\mathrm {all\,space} }e^{-i\mathbf {p} \cdot \mathbf {r} /\hbar }\Psi (\mathbf {r} ,s_{z},t)\mathrm {d} ^{3}\mathbf {r} \\&\upharpoonleft \downharpoonright \\\Psi (\mathbf {r} ,s_{z},t)&={\frac {1}{{\sqrt {2\pi \hbar }}^{3}}}\int \limits _{\mathrm {all\,space} }e^{+i\mathbf {p} \cdot \mathbf {r} /\hbar }\Phi (\mathbf {p} ,s_{z},t)\mathrm {d} ^{3}\mathbf {p} _{n}\\\end{aligned}}$
General probability distribution	V_j = volume (3d region) particle may occupy, P = Probability that particle 1 has position r₁ in volume V₁ with spin s_z1 and particle 2 has position r₂ in volume V₂ with spin s_z2, etc.	$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int _{V_{N}}\cdots \int _{V_{2}}\int _{V_{1}}\left\|\Psi \right\|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,\!$
General normalization condition		$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int \limits _{\mathrm {all\,space} }\cdots \int \limits _{\mathrm {all\,space} }\;\int \limits _{\mathrm {all\,space} }\left\|\Psi \right\|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}=1\,\!$

Property or effect	Nomenclature	Equation
Planck–Einstein equation and de Broglie wavelength relations	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	$\mathbf {P} =(E/c,\mathbf {p} )=\hbar (\omega /c,\mathbf {k} )=\hbar \mathbf {K}$
Schrödinger equation	Ψ = wavefunction of the system Ĥ = Hamiltonian operator, E = energy eigenvalue of system i is the imaginary unit t = time	General time-dependent case: $i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi$ Time-independent case: ${\hat {H}}\Psi =E\Psi$
Heisenberg equation	Â = operator of an observable property [ ] is the commutator $\langle \,\rangle$ denotes the average	${\frac {d}{dt}}{\hat {A}}(t)={\frac {i}{\hbar }}[{\hat {H}},{\hat {A}}(t)]+{\frac {\partial {\hat {A}}(t)}{\partial t}}$
Time evolution in Heisenberg picture (Ehrenfest theorem)	m = mass, V = potential energy, r = position, p = momentum, of a particle.	${\frac {d}{dt}}\langle {\hat {A}}\rangle ={\frac {1}{i\hbar }}\langle [{\hat {A}},{\hat {H}}]\rangle +\left\langle {\frac {\partial {\hat {A}}}{\partial t}}\right\rangle$ For momentum and position; $m{\frac {d}{dt}}\langle \mathbf {r} \rangle =\langle \mathbf {p} \rangle$ ${\frac {d}{dt}}\langle \mathbf {p} \rangle =-\langle \nabla V\rangle$

	One particle	N particles
One dimension	${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x)=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N})\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N})\end{aligned}}$ where the position of particle n is x_n.
	$E\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi +V\Psi$	$E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi (x,t)=\psi (x)e^{-iEt/\hbar }\,.$ 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):^[1] $\\|\psi \\|^{2}=\int \|\psi (x)\|^{2}\,dx.\,$	$\Psi =e^{-iEt/\hbar }\psi (x_{1},x_{2}\cdots x_{N})$ for non-interacting particles $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (x_{n})\,,\quad V(x_{1},x_{2},\cdots x_{N})=\sum _{n=1}^{N}V(x_{n})\,.$
Three dimensions	${\begin{aligned}{\hat {H}}&={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} )\\&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\end{aligned}}$ where the position of the particle is r = (x, y, z).	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {\mathbf {p} }}_{n}\cdot {\hat {\mathbf {p} }}_{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})\end{aligned}}$ where the position of particle n is r _n = (x_n, y_n, z_n), and the Laplacian for particle n using the corresponding position coordinates is $\nabla _{n}^{2}={\frac {\partial ^{2}}{{\partial x_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial y_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial z_{n}}^{2}}}$
	$E\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi$	$E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi$
	$\Psi =\psi (\mathbf {r} )e^{-iEt/\hbar }$	$\Psi =e^{-iEt/\hbar }\psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N})$ for non-interacting particles $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (\mathbf {r} _{n})\,,\quad V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})=\sum _{n=1}^{N}V(\mathbf {r} _{n})$

	One particle	N particles
One dimension	${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N},t)\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N},t)\end{aligned}}$ where the position of particle n is x_n.
	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi +V\Psi$	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi =\Psi (x,t)$	$\Psi =\Psi (x_{1},x_{2}\cdots x_{N},t)$
Three dimensions	${\begin{aligned}{\hat {H}}&={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} ,t)\\&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\\\end{aligned}}$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {\mathbf {p} }}_{n}\cdot {\hat {\mathbf {p} }}_{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)\end{aligned}}$
	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi$	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi$ This last equation is in a very high dimension,^[2] so the solutions are not easy to visualize.
	$\Psi =\Psi (\mathbf {r} ,t)$	$\Psi =\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)$

List of equations in quantum mechanics

Wavefunctions

Equations

Wave–particle duality and time evolution

Non-relativistic time-independent Schrödinger equation

Non-relativistic time-dependent Schrödinger equation

Photoemission

Quantum uncertainty

Angular momentum

Hydrogen atom

See also

Footnotes

Sources

Further reading

Related Articles

Property/Effect	Nomenclature	Equation
Photoelectric equation	K_max = 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)	$K_{\mathrm {max} }=hf-\Phi \,\!$
Threshold frequency and Work function	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: $\phi =hf_{0}\,\!$
Photon momentum	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: $p=hf/c=h/\lambda \,\!$

Property or effect	Nomenclature	Equation
Heisenberg's uncertainty principles	n = number of photons φ = wave phase [, ] = commutator	Position–momentum $\sigma (x)\sigma (p)\geq {\frac {\hbar }{2}}\,\!$ Energy-time $\sigma (E)\sigma (t)\geq {\frac {\hbar }{2}}\,\!$ Number-phase $\sigma (n)\sigma (\phi )\geq {\frac {\hbar }{2}}\,\!$
Dispersion of observable	A = observables (eigenvalues of operator)	${\begin{aligned}\sigma (A)^{2}&=\langle (A-\langle A\rangle )^{2}\rangle \\&=\langle A^{2}\rangle -\langle A\rangle ^{2}\end{aligned}}$
General uncertainty relation	A, B = observables (eigenvalues of operator)	$\sigma (A)\sigma (B)\geq {\frac {1}{2}}\langle i[{\hat {A}},{\hat {B}}]\rangle$

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 P(E_i) = probability of energy E_i g(E_i) = degeneracy of energy E_i (no of states with same energy) μ = chemical potential
Bose–Einstein distribution (bosons)	$P(E_{i})={\frac {g(E_{i})}{e^{(E_{i}-\mu )/kT}-1}}$

Property or effect	Nomenclature	Equation
Angular momentum quantum numbers	s = spin quantum number m_s = spin magnetic quantum number ℓ = Azimuthal quantum number m_ℓ = azimuthal magnetic quantum number j = total angular momentum quantum number m_j = total angular momentum magnetic quantum number	Spin: ${\begin{aligned}&\Vert \mathbf {s} \Vert ={\sqrt {s\,(s+1)}}\,\hbar \\&m_{s}\in \{-s,-s+1\cdots s-1,s\}\\\end{aligned}}\,\!$ Orbital: ${\begin{aligned}&\ell \in \{0\cdots n-1\}\\&m_{\ell }\in \{-\ell ,-\ell +1\cdots \ell -1,\ell \}\\\end{aligned}}\,\!$ Total: ${\begin{aligned}&j=\ell +s\\&m_{j}\in \{\|\ell -s\|,\|\ell -s\|+1\cdots \|\ell +s\|-1,\|\ell +s\|\}\\\end{aligned}}\,\!$
Angular momentum magnitudes	angular momementa: S = Spin, L = orbital, J = total	Spin magnitude: $\|\mathbf {S} \|=\hbar {\sqrt {s(s+1)}}\,\!$ Orbital magnitude: $\|\mathbf {L} \|=\hbar {\sqrt {\ell (\ell +1)}}\,\!$ Total magnitude: $\mathbf {J} =\mathbf {L} +\mathbf {S} \,\!$ $\|\mathbf {J} \|=\hbar {\sqrt {j(j+1)}}\,\!$
Angular momentum components		Spin: $S_{z}=m_{s}\hbar \,\!$ Orbital: $L_{z}=m_{\ell }\hbar \,\!$

Property or effect	Nomenclature	Equation
orbital magnetic dipole moment	e = electron charge m_e = electron rest mass L = electron orbital angular momentum g_ℓ = orbital Landé g-factor μ_B = Bohr magneton	${\boldsymbol {\mu }}_{\ell }=-e\mathbf {L} /2m_{e}=g_{\ell }{\frac {\mu _{B}}{\hbar }}\mathbf {L} \,\!$ z-component: $\mu _{\ell ,z}=-m_{\ell }\mu _{B}\,\!$
spin magnetic dipole moment	S = electron spin angular momentum g_s = spin Landé g-factor	${\boldsymbol {\mu }}_{s}=-e\mathbf {S} /m_{e}=g_{s}{\frac {\mu _{B}}{\hbar }}\mathbf {S} \,\!$ z-component: $\mu _{s,z}=-eS_{z}/m_{e}=g_{s}eS_{z}/2m_{e}\,\!$
dipole moment potential	U = potential energy of dipole in field	$U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!$

Property or effect	Nomenclature	Equation
Energy level	E_n = energy eigenvalue n = principal quantum number e = electron charge m_e = electron rest mass ε₀ = permittivity of free space h = Planck constant	$E_{n}=-me^{4}/8\varepsilon _{0}^{2}h^{2}n^{2}=-13.61\,\mathrm {eV} /n^{2}$
Spectrum	λ = wavelength of emitted photon, during electronic transition from E_i to E_j	${\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}<n_{i}\,\!$