# List of equations in quantum mechanics

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 ħ = h/2π, 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 | $\rho =\left|\Psi \right|^{2}=\Psi ^{*}\Psi$ | 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]$^{−2}$ |
The general form of wavefunction for a system of particles, each with position r$_i$ and z-component of spin s$_{zi}$. 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 | r = (r$_1$, r$_2$... r$_N$) s$_z$ = (s$_{z1}$, s$_{z2}$, ..., s$_{zN}$) | 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 r$_1$ in volume V$_1$ with spin s$_{z1}$ and particle 2 has position r$_2$ in volume V$_2$ 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 | | $P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int \limits _{\mathrm {all\,space} }\cdots \int \limits _{\mathrm {all\,space} }\;\int \limits _{\mathrm {all\,space} }\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}=1\,\!$ |

## Equations
### Wave–particle duality and time evolution

| 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; $m{\frac {d}{dt}}\langle \mathbf {r} \rangle =\langle \mathbf {p} \rangle$ ${\frac {d}{dt}}\langle \mathbf {p} \rangle =-\langle \nabla V\rangle$ |

#### 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 | ${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x)=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)$ | ${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N})\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N})\end{aligned}}$ where the position of particle n is x$_n$. |
| One dimension | $E\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi +V\Psi$ | $E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$ |
| One dimension | $\Psi (x,t)=\psi (x)e^{-iEt/\hbar }\,.$ There is a further restriction — the solution must not grow at infinity, so that it has either a finite L$^2$-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): $\|\psi \|^{2}=\int |\psi (x)|^{2}\,dx.\,$ | $\Psi =e^{-iEt/\hbar }\psi (x_{1},x_{2}\cdots x_{N})$ 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 $\nabla _{n}^{2}={\frac {\partial ^{2}}{{\partial x_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial y_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial z_{n}}^{2}}}$ |
| Three dimensions | $E\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi$ | $E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi$ |
| Three dimensions | $\Psi =\psi (\mathbf {r} )e^{-iEt/\hbar }$ | $\Psi =e^{-iEt/\hbar }\psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N})$ 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

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

### Photoemission

| 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) f$_0$, ν$_0$ = 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

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

### Angular momentum

| 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$_ℓ$ = 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 | $U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!$ |

### Hydrogen atom

| Property or effect | Nomenclature | Equation |
| Energy level | E$_n$ = energy eigenvalue n = principal quantum number e = electron charge m$_e$ = electron rest mass ε$_0$ = 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$_j$ | ${\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}<n_{i}\,\!$ |