View source for List of equations in quantum mechanics

{{short description|None}}

{{quantum mechanics}}
This article summarizes [[equation]]s in the theory of [[quantum mechanics]].

==Wavefunctions==

A fundamental [[physical constant]] occurring in quantum mechanics is the '''[[Planck constant]]''', ''h''. A common abbreviation is {{nowrap|1=''ħ'' = ''h''/2''π''}}, also known as the ''reduced Planck constant'' or ''Dirac constant''.

{| class="wikitable"
|-
! scope="col" width="130" | Quantity (Common Name/s) 
! scope="col" width="100" | (Common) Symbol/s
! scope="col" width="150" | Defining Equation
! scope="col" width="100" | SI Units 
! scope="col" width="100" | Dimension
|-
|[[Wave function|Wavefunction]]
| ''ψ, Ψ'' 
| To solve from the [[Schrödinger equation]]
| varies with situation and number of particles
| 
|-
| Wavefunction [[probability density function|probability density]]
| ''ρ''
|<math>\rho = \left | \Psi \right |^2 = \Psi^* \Psi </math>
| m<sup>−3</sup>
| [L]<sup>−3</sup>
|-
|Wavefunction [[probability current]]
| '''j'''
|Non-relativistic, no external field:
<math>\begin{align}
\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{align}</math>

star * is [[complex conjugate]]
| m<sup>−2</sup> s<sup>−1</sup>
| [T]<sup>−1</sup> [L]<sup>−2</sup>
|-
|}

The general form of [[wavefunction]] for a system of particles, each with position '''r'''<sub>''i''</sub> and z-component of spin ''s<sub>z i</sub>''. Sums are over the discrete variable ''s<sub>z</sub>'', 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.

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
! [[Wavefunction]] for ''N'' particles in 3d
| {{plainlist}}
*'''r''' = ('''r'''<sub>1</sub>, '''r'''<sub>2</sub>... '''r'''<sub>''N''</sub>)
*'''s<sub>z</sub>''' = (''s''<sub>''z'' 1</sub>, ''s''<sub>''z'' 2</sub>, ..., ''s<sub>z N</sub>'')
{{endplainlist}}
| In function notation:

<math> \Psi  = \Psi \left (\mathbf{r}, \mathbf{s_z}, t \right ) </math>

in [[bra–ket notation]]:
<math> |\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 </math>

for non-interacting particles:

<math> \Psi  = \prod_{n=1}^N\Psi \left (\mathbf{r}_n,s_{zn}, t \right ) </math>
|-
!Position-momentum Fourier transform (1 particle in 3d)
| {{plainlist}}
*Φ = momentum-space wavefunction
*Ψ = position-space wavefunction
{{endplainlist}}
|<math>\begin{align} \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{align}</math>
|-
!General probability distribution
| {{plainlist}}
*''V<sub>j</sub>'' = volume (3d region) particle may occupy,
*''P'' = Probability that particle 1 has position '''r'''<sub>1</sub> in volume ''V''<sub>1</sub> with spin ''s''<sub>''z''1</sub> '''and''' particle 2 has position '''r'''<sub>2</sub> in volume ''V''<sub>2</sub> with spin ''s''<sub>''z''2</sub>, etc.
{{endplainlist}}
|<math> 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\,\!</math>
|-
!General [[Normalizable wave function|normalization]] condition
|
|<math> 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\,\!</math>
|-
|}

==Equations==

===Wave–particle duality and time evolution===

{| class="wikitable"
|-
! scope="col" width="150" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Planck–Einstein equation]] and [[de Broglie wavelength]] relations
|{{plainlist}}
*'''P''' = (''E/c'', '''p''') is the [[four-momentum]],
*'''K''' = (ω/''c'', '''k''') is the [[four-vector#Four-wavevector|four-wavevector]],
*''E'' = energy of particle
*ω = 2π''f'' is the angular frequency and frequency of the particle
*''ħ'' = ''h''/2π are the [[Planck constant]]s
*''c'' = [[speed of light]]
{{endplainlist}}
|<math>\mathbf{P} = (E/c, \mathbf{p}) = \hbar(\omega /c ,\mathbf{k}) = \hbar \mathbf{K}</math>
|-
![[Schrödinger equation]]
|{{plainlist}}
*''Ψ'' = [[wavefunction]] of the system
*''Ĥ'' = [[Hamiltonian (quantum mechanics)|Hamiltonian operator]],
*''E'' = energy eigenvalue of system
*''i'' is the [[imaginary unit]]
*''t'' = time
{{endplainlist}}
|General time-dependent case:
<math>i\hbar\frac{\partial}{\partial t} \Psi = \hat{H}\Psi</math>

Time-independent case:
<math>\hat{H}\Psi = E\Psi</math>
|-
![[Heisenberg picture|Heisenberg equation]]
|{{plainlist}}
*''Â'' = operator of an observable property
*[ ] is the [[commutator]]
*<math>\langle \, \rangle</math> denotes the average
{{endplainlist}}
|<math>\frac{d}{dt}\hat{A}(t)=\frac{i}{\hbar}[\hat{H},\hat{A}(t)]+\frac{\partial \hat{A}(t)}{\partial t}</math>
|-
!Time evolution in Heisenberg picture ([[Ehrenfest theorem]])
|{{plainlist}}
*''m'' = [[mass]],
*''V'' = [[potential energy]], 
*'''r''' = [[position space|position]],
*'''p''' = [[momentum]],
{{endplainlist}}
of a particle.
|<math>\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</math>

For momentum and position;

<math>m\frac{d}{dt}\langle \mathbf{r}\rangle = \langle \mathbf{p} \rangle</math>

<math>\frac{d}{dt}\langle \mathbf{p}\rangle = -\langle \nabla V \rangle</math>
|-

|-
|}

====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]].

{| class="wikitable"
|-
| scope="col" style="width:10px;"|
| scope="col" style="width:450px;"| '''One particle''' 
| scope="col" style="width:450px;"| ''' ''N'' particles'''
|-
|rowspan="3"| '''One dimension''' 
| <math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x) = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x)</math>
| <math> \begin{align}
\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{align}</math>

where the position of particle ''n'' is ''x<sub>n</sub>''.
|-
| <math> E\Psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\Psi + V\Psi </math>
| <math> E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, .</math>
|-
|<math> \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . </math>

There is a further restriction — the solution must not grow at infinity, so that it has either a finite '''''L''<sup>2</sup>-norm''' (if it is a [[bound state]]) or a slowly diverging norm (if it is part of a [[continuum (theory)|continuum]]):<ref>
{{cite book
 |last1=Feynman |first1=R.P.
 |last2=Leighton |first2=R.B.
 |last3=Sand |first3=M.
 |year=1964
 |chapter=Operators
 |title=[[The Feynman Lectures on Physics]]
 |volume=3 |pages=20–7
 |publisher=[[Addison-Wesley]]
 |isbn=0-201-02115-3
}}</ref>
<math>\| \psi \|^2 = \int |\psi(x)|^2\, dx.\,</math>
| <math> \Psi = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N) </math>

for non-interacting particles

<math> \Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, , \quad V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, .</math>
|-
|rowspan="3"| '''Three dimensions''' 
| <math> \begin{align}\hat{H} & = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \\
& = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) 
\end{align}</math>

where the position of the particle is '''r''' = (''x, y, z'').
| <math> \begin{align} \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{align}</math>

where the position of particle ''n'' is '''r''' ''<sub>n</sub>'' = (''x<sub>n</sub>'', ''y<sub>n</sub>'', ''z<sub>n</sub>''), and the Laplacian for particle ''n'' using the corresponding position coordinates is

<math>\nabla_n^2=\frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}</math>
|-
| <math> E\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi </math>
| <math> E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi </math>
|-
|<math> \Psi = \psi(\mathbf{r}) e^{-iEt/\hbar} </math>
| <math> \Psi = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N) </math>

for non-interacting particles

<math> \Psi = e^{-i{E t/\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) </math>
|}

====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.

{| class="wikitable"
|-
| scope="col" style="width:10px;"|
| scope="col" style="width:450px;"| '''One particle''' 
| scope="col" style="width:450px;"| ''' ''N'' particles'''
|-
|rowspan="3"| '''One dimension''' 
| <math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)</math>
| <math> \begin{align}
\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{align} 
</math>

where the position of particle ''n'' is ''x<sub>n</sub>''.
|-
| <math> i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi + V\Psi </math>
| <math> 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 \, .</math>
|-
| <math> \Psi = \Psi(x,t) </math>
| <math> \Psi = \Psi(x_1,x_2\cdots x_N,t) </math>
|-
|rowspan="3"| '''Three dimensions''' 
| <math> \begin{align}\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{align}</math>
| <math> \begin{align} \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{align}</math> 
|-
| <math> i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi </math>
| <math> 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 </math>

This last equation is in a very high dimension,<ref>
{{cite book
 |last=Shankar |first=R.
 |year=1994
 |title=Principles of Quantum Mechanics
 |url=https://archive.org/details/principlesquantu00shan_139 |url-access=limited |page=[https://archive.org/details/principlesquantu00shan_139/page/n160 141]
 |publisher=[[Kluwer Academic]]/[[Plenum Publishers]]
 |isbn=978-0-306-44790-7
}}</ref> so the solutions are not easy to visualize. 
|-
| <math> \Psi = \Psi(\mathbf{r},t) </math>
| <math> \Psi = \Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) </math>
|}


===Photoemission===

{| class="wikitable"
!Property/Effect
!Nomenclature
!Equation
|-
![[Photoelectric]] equation
|{{plainlist}}
*''K''<sub>max</sub> = Maximum kinetic energy of ejected electron (J)
*''h'' = [[Planck's constant]]
*''f'' = frequency of incident photons (Hz = s<sup>−1</sup>)
*''φ'', Φ = [[Work function]] of the material the photons are incident on (J)
{{endplainlist}}
|<math>K_\mathrm{max} = hf - \Phi\,\!</math>
|-
![[Photoelectric effect|Threshold frequency]] and [[Photoelectric effect|Work function]]
| {{plainlist}}
*''φ'', Φ = Work function of the material the photons are incident on (J)
*''f''<sub>0</sub>, ''ν''<sub>0</sub> = Threshold frequency (Hz = s<sup>−1</sup>)
{{endplainlist}}
| Can only be found by experiment.

The De Broglie relations give the relation between them:

<math>\phi = hf_0\,\!</math>
|-
![[Photon]] momentum
| {{plainlist}}
*''p'' = momentum of photon (kg m s<sup>−1</sup>)
*''f'' = frequency of photon (Hz = s<sup>−1</sup>)
*''λ'' = wavelength of photon (m)
{{endplainlist}}
|
The De Broglie relations give:

<math>p = hf/c = h/\lambda\,\!</math>
|-
|}

===Quantum uncertainty===

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Heisenberg's uncertainty principle]]s
|{{plainlist}}
*''n'' = number of photons
*''φ'' = wave phase
*[, ] = [[commutator]]
{{endplainlist}}

|Position-momentum

<math>\sigma(x) \sigma(p) \ge \frac{\hbar}{2} \,\!</math>

Energy-time
<math>\sigma(E) \sigma(t) \ge \frac{\hbar}{2} \,\!</math>

Number-phase
<math>\sigma(n) \sigma(\phi) \ge \frac{\hbar}{2} \,\!</math>
|-
!Dispersion of observable
|''A'' = observables (eigenvalues of operator)
|
<math>\begin{align}
\sigma(A)^2 & = \langle(A-\langle A \rangle)^2\rangle \\
& = \langle A^2 \rangle - \langle A \rangle^2
\end{align}</math>
|-
!General uncertainty relation
|''A'', ''B'' = observables (eigenvalues of operator)
|<math>\sigma(A)\sigma(B) \geq \frac{1}{2}\langle i[\hat{A}, \hat{B}] \rangle</math>
|-
|}

{| class="wikitable"
|+Probability Distributions
! scope="col" width="100" | Property or effect 
! scope="col" width="10" | Equation 
|-
![[Density of states]]
|<math>N(E) = 8\sqrt{2}\pi m^{3/2}E^{1/2}/h^3\,\!</math>
|-
! [[Fermi–Dirac distribution]] (fermions)
|<math>P(E_i) = \frac{g(E_i)}{e^{(E-\mu)/kT} + 1}</math>
where
{{plainlist}}
*''P''(''E<sub>i</sub>'') = probability of energy ''E<sub>i</sub>''
*''g''(''E<sub>i</sub>'') = degeneracy of energy ''E<sub>i</sub>'' (no of states with same energy)
*μ = chemical potential
{{endplainlist}}
|-
! [[Bose–Einstein distribution]] (bosons) 
|<math>P(E_i) = \frac{g(E_i)}{e^{(E_i-\mu)/kT}-1}</math>
|-
|}

===Angular momentum===

{{main|angular momentum operator|quantum number}}

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Angular momentum]] [[quantum number]]s
|{{plainlist}}
*''s'' = [[spin quantum number]]
*''m<sub>s</sub>'' = spin magnetic quantum number
*''ℓ'' = [[Azimuthal quantum number]]
*''m<sub>ℓ</sub>'' = azimuthal magnetic quantum number
*''j'' = total angular momentum quantum number
*''m<sub>j</sub>'' = total angular momentum magnetic quantum number
{{endplainlist}}

|
Spin:
<math>\begin{align}& \Vert \mathbf{s} \Vert = \sqrt{s \, (s+1)} \, \hbar \\
& m_s \in \{-s,-s+1\cdots s-1,s\}\\
\end{align}\,\!</math>

Orbital:
<math>\begin{align}& \ell \in \{0 \cdots n-1\} \\
& m_\ell \in \{-\ell,-\ell+1\cdots \ell-1,\ell\}\\
\end{align}\,\!</math>

Total:
<math>\begin{align}& j = \ell +s \\
& m_j \in \{|\ell-s|,|\ell-s|+1 \cdots |\ell+s|-1,|\ell+s| \} \\
\end{align}\,\!</math>
|-
![[Angular momentum]] magnitudes
|angular momementa:{{plainlist}}
*''S'' = Spin,
*''L'' = orbital,
*''J'' = total
{{endplainlist}}

|Spin magnitude:
<math>|\mathbf{S}| = \hbar\sqrt{s(s+1)}\,\!</math>

Orbital magnitude:
<math>|\mathbf{L}| = \hbar\sqrt{\ell(\ell+1)}\,\!</math>

Total magnitude:
<math>\mathbf{J} = \mathbf{L} + \mathbf{S}\,\!</math>

<math>|\mathbf{J}| = \hbar\sqrt{j(j+1)}\,\!</math>
|-
![[Angular momentum]] components
|
|Spin:
<math>S_z = m_s \hbar\,\!</math>

Orbital:
<math>L_z = m_\ell \hbar\,\!</math>
|-
|}

;Magnetic moments

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

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Electron magnetic dipole moment#Orbital magnetic dipole moment|orbital magnetic dipole moment]]
|{{plainlist}}
*''e'' = [[electron charge]]
*''m<sub>e</sub>'' = electron rest mass
*'''L''' = electron orbital angular momentum
*''g<sub>{{ell}}</sub>'' = orbital [[Landé g-factor]]
*μ<sub>''B''</sub> = [[Bohr magneton]]
{{endplainlist}}
|<math>\boldsymbol{\mu}_\ell = -e\mathbf{L}/2m_e = g_\ell \frac{\mu_B}{\hbar} \mathbf{L}\,\!</math>
z-component:
<math>\mu_{\ell,z} = -m_\ell\mu_B\,\!</math>
|-
![[Electron magnetic dipole moment#Orbital magnetic dipole moment|spin magnetic dipole moment]]
|{{plainlist}}
*'''S''' = electron spin angular momentum
*''g<sub>s</sub>'' = spin [[Landé g-factor]]
{{endplainlist}}
|<math>\boldsymbol{\mu}_s = -e\mathbf{S}/m_e = g_s \frac{\mu_B}{\hbar} \mathbf{S}\,\!</math>

z-component:
<math>\mu_{s,z} = -e S_z/m_e = g_seS_z/2m_e\,\!</math>
|-
![[magnetic moment|dipole moment]] potential
|''U'' = potential energy of dipole in field
|<math>U = -\boldsymbol{\mu}\cdot\mathbf{B} = -\mu_z B\,\!</math>
|-
|}

===The Hydrogen atom===
{{main|Hydrogen atom}}

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
! Energy level
|{{plainlist}}
*''E<sub>n</sub>'' = energy [[eigenvalue]]
*''n'' = [[principal quantum number]]
*''e'' = [[electron charge]]
*''m<sub>e</sub>'' = electron rest mass
*ε<sub>0</sub> = permittivity of free space
*''h'' = [[Planck's constant]]
{{endplainlist}}
|<math>E_n = -m e^4 / 8\varepsilon_0^2 h^2 n^2 = -13.61\,\mathrm{eV}/n^2</math>
|-
! Spectrum
|λ = wavelength of emitted photon, during [[Excited state|electronic transition]] from ''E<sub>i</sub>'' to ''E<sub>j</sub>'' 
|<math>\frac{1}{\lambda} = R\left(\frac{1}{n_j^2} - \frac{1}{n_i^2}\right), \, n_j<n_i\,\!</math>
|-
|}

==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]]

==Footnotes==

{{Reflist}}

==Sources==

* {{cite book|author1=P.M. Whelan |author2=M.J. Hodgeson | title=Essential Principles of Physics| publisher=John Murray|edition=2nd| year=1978 | isbn=0-7195-3382-1}}
* {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| publisher=Cambridge University Press| year=2010| isbn=978-0-521-57507-2| url-access=registration| url=https://archive.org/details/cambridgehandboo0000woan}}
* {{cite book| author=A. Halpern| title=3000 Solved Problems in Physics, Schaum Series| publisher=Mc Graw Hill| year=1988| isbn=978-0-07-025734-4}}
* {{cite book|pages=12–13|author1=R. G. Lerner|author1-link=Rita G. Lerner |author2=G. L. Trigg | title=Encyclopaedia of Physics| publisher=VHC Publishers, Hans Warlimont, Springer|edition=2nd| year=2005| isbn=978-0-07-025734-4}}
* {{cite book| author=C. B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| edition=2nd| year=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park}}
* {{cite book|author1=P. A. Tipler |author2=G. Mosca | title=Physics for Scientists and Engineers: With Modern Physics| publisher=W. H. Freeman and Co|edition=6th| year=2008| isbn=978-1-4292-0265-7}}
* {{cite book|title=Analytical Mechanics|author1=L.N. Hand |author2=J. D. Finch |publisher=Cambridge University Press |year=2008|isbn=978-0-521-57572-0}}
* {{cite book|title=Mechanics, Vibrations and Waves|author1=T. B. Arkill |author2=C. J. Millar |publisher=John Murray |year=1974|isbn=0-7195-2882-8}}
* {{cite book|title=The Physics of Vibrations and Waves|edition=3rd|author=H.J. Pain|publisher=John Wiley & Sons |year=1983|isbn=0-471-90182-2}}
* {{cite book|title=Dynamics and Relativity|author1=J. R. Forshaw |author2=A. G. Smith |publisher=Wiley |year=2009|isbn=978-0-470-01460-8}}
* {{cite book|title=Electricity and Modern Physics |edition=2nd|author=G. A. G. Bennet|publisher=Edward Arnold (UK)|year=1974|isbn=0-7131-2459-8}}
* {{cite book|title=Electromagnetism|author1=I. S. Grant |author2=W. R. Phillips |author3=Manchester Physics |publisher=John Wiley & Sons|year=2008|isbn=978-0-471-92712-9|edition=2nd }}
* {{cite book|title=Introduction to Electrodynamics|edition=3rd |author=D.J. Griffiths|publisher=Pearson Education, Dorling Kindersley |year=2007|isbn=978-81-7758-293-2}}

==Further reading==

* {{cite book|title=Physics with Modern Applications|author=L. H. Greenberg|publisher=Holt-Saunders International W. B. Saunders and Co|year=1978|isbn=0-7216-4247-0|url-access=registration|url=https://archive.org/details/physicswithmoder0000gree}}
* {{cite book|title=Principles of Physics|author1=J. B. Marion |author2=W. F. Hornyak |publisher=Holt-Saunders International Saunders College|year=1984|isbn=4-8337-0195-2}}
* {{cite book|title=Concepts of Modern Physics|edition=4th|author=A. Beiser|publisher=McGraw-Hill (International)|year=1987|isbn=0-07-100144-1}}
* {{cite book|title=University Physics – With Modern Physics|edition=12th|author1=H. D. Young |author2=R. A. Freedman |publisher=Addison-Wesley (Pearson International)|year=2008|isbn=978-0-321-50130-1}}

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