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List of equations in quantum mechanics

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Quantum mechanics
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$i\hbar {\frac {d}{dt}}\|\Psi \rangle ={\hat {H}}\|\Psi \rangle$ Schrödinger equation
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v t e

This article summarizesequations in the theory ofquantum mechanics.

Wavefunctions

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A fundamentalphysical constant occurring in quantum mechanics is thePlanck constant,h. A common abbreviation isħ =h/2π, also known as thereduced Planck constant orDirac constant.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
Wavefunction	ψ, Ψ	To solve from theSchrödinger equation	varies with situation and number of particles
Wavefunctionprobability density	ρ	$\rho =\left\|\Psi \right\|^{2}=\Psi ^{*}\Psi$	m⁻³	[L]⁻³
Wavefunctionprobability 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 * iscomplex conjugate	m⁻²⋅s⁻¹	[T]⁻¹ [L]⁻²

Quantity (common name/s)

(Common) symbol/s

Defining equation

SI unit

Dimension

Wavefunction

ψ, Ψ

To solve from theSchrödinger equation

varies with situation and number of particles

Wavefunctionprobability density

\rho =\left|\Psi \right|^{2}=\Psi ^{*}\Psi

m⁻³

[L]⁻³

Wavefunctionprobability current

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 * iscomplex conjugate

m⁻²⋅s⁻¹

[T]⁻¹ [L]⁻²

The general form ofwavefunction for a system of particles, each with positionr_i and z-component of spins_{z i}. Sums are over the discrete variables_z, integrals over continuous positionsr.

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 forN particles in 3d	r = (r₁,r₂...r_N) s_z = (s_{z 1},s_{z 2}, ...,s_{z N})	In function notation: $\Psi =\Psi \left(\mathbf {r} ,\mathbf {s_{z}} ,t\right)$ inbra–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 positionr₁ in volumeV₁ with spins_z1and particle 2 has positionr₂ in volumeV₂ with spins_z2, etc.	$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int _{V_{N}}\cdots \int _{V_{2}}\int _{V_{1}}\left\|\Psi \right\|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,\!$
Generalnormalization condition		$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int \limits _{\mathrm {all\,space} }\cdots \int \limits _{\mathrm {all\,space} }\;\int \limits _{\mathrm {all\,space} }\left\|\Psi \right\|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}=1\,\!$

Equations

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Wave–particle duality and time evolution

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Property or effect	Nomenclature	Equation
Planck–Einstein equation andde Broglie wavelength relations	P = (E/c,p) is thefour-momentum, K = (ω/c,k) is thefour-wavevector, E = energy of particle ω = 2πf is the angular frequency and frequency of the particle ħ =h/2π are thePlanck 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 theimaginary 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 thecommutator $\langle \,\rangle$ 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; $m{\frac {d}{dt}}\langle \mathbf {r} \rangle =\langle \mathbf {p} \rangle$ ${\frac {d}{dt}}\langle \mathbf {p} \rangle =-\langle \nabla V\rangle$

Non-relativistic time-independent Schrödinger equation

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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, thepartial derivative reduces to anordinary derivative.

	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 particlen isx_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 finiteL²-norm (if it is abound state) or a slowly diverging norm (if it is part of acontinuum):^[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 isr = (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 particlen isr_n = (x_n,y_n,z_n), and the Laplacian for particlen 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 $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (\mathbf {r} _{n})\,,\quad V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})=\sum _{n=1}^{N}V(\mathbf {r} _{n})$

Non-relativistic time-dependent Schrödinger equation

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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	${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N},t)\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N},t)\end{aligned}}$ where the position of particlen isx_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.
	$\Psi =\Psi (\mathbf {r} ,t)$	$\Psi =\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)$

Photoemission

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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 andWork 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/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 andWork 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:

$p=hf/c=h/\lambda \,\!$

Quantum uncertainty

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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

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

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

Angular momentum

[edit]

Main articles:angular momentum operator andquantum number

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

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: $L_{z}=m_{\ell }\hbar \,\!$

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	e =electron charge m_e = electron rest mass L = electron orbital angular momentum g_ℓ = orbitalLandé 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 = spinLandé 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	$U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!$

Hydrogen atom

[edit]

Main article:Hydrogen atom

Property or effect	Nomenclature	Equation
Energy level	E_n = energyeigenvalue 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, duringelectronic transition fromE_i toE_j	${\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}<n_{i}\,\!$

Footnotes

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^Feynman, R.P.; Leighton, R.B.; Sand, M. (1964). "Operators".The Feynman Lectures on Physics. Vol. 3.Addison-Wesley. pp. 20–7.ISBN 0-201-02115-3.{{cite book}}:ISBN / Date incompatibility (help)
^Shankar, R. (1994).Principles of Quantum Mechanics.Kluwer Academic/Plenum Publishers. p. 141.ISBN 978-0-306-44790-7.

Sources

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v t e SI units
Base units	ampere candela kelvin kilogram metre mole second
Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber
Other accepted units	astronomical unit dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne
See also	Conversion of units Metric prefixes Historical definitions of the SI base units 2019 revision of the SI System of units of measurement
Category

Movatterモバイル変換

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