List of equations in 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 Units Dimension
Wavefunction ψ, Ψ To solve from the Schrödinger equation varies with situation and number of particles
Wavefunction probability density ρ ${\displaystyle \rho =\left|\Psi \right|^{2}=\Psi ^{*}\Psi }$  m−3 [L]−3
Wavefunction probability current j Non-relativistic, no external field:

{\displaystyle {\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 ri and z-component of spin sz i. Sums are over the discrete variable sz, 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 = (r1, r2... rN)
• sz = (sz 1, sz 2, ..., sz N)
In function notation:

${\displaystyle \Psi =\Psi \left(\mathbf {r} ,\mathbf {s_{z}} ,t\right)}$

in bra–ket notation: ${\displaystyle |\Psi \rangle =\sum _{s_{z1}}\sum _{s_{z2}}\cdots \sum _{s_{zN}}\int _{V_{1}}\int _{V_{2}}\cdots \int _{V_{N}}\mathrm {d} \mathbf {r} _{1}\mathrm {d} \mathbf {r} _{2}\cdots \mathrm {d} \mathbf {r} _{N}\Psi |\mathbf {r} ,\mathbf {s_{z}} \rangle }$

for non-interacting particles:

${\displaystyle \Psi =\prod _{n=1}^{N}\Psi \left(\mathbf {r} _{n},s_{zn},t\right)}$

Position-momentum Fourier transform (1 particle in 3d)
• Φ = momentum-space wavefunction
• Ψ = position-space wavefunction
{\displaystyle {\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
• 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.
${\displaystyle P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int _{V_{N}}\cdots \int _{V_{2}}\int _{V_{1}}\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,\!}$
General normalization condition ${\displaystyle P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int \limits _{\mathrm {all\,space} }\cdots \int \limits _{\mathrm {all\,space} }\;\int \limits _{\mathrm {all\,space} }\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}=1\,\!}$

Equations

Wave–particle duality and time evolution

Property or effect Nomenclature Equation
Planck–Einstein equation and de Broglie wavelength relations
${\displaystyle \mathbf {P} =(E/c,\mathbf {p} )=\hbar (\omega /c,\mathbf {k} )=\hbar \mathbf {K} }$
Schrödinger equation
General time-dependent case:

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }$

Time-independent case: ${\displaystyle {\hat {H}}\Psi =E\Psi }$

Heisenberg equation
• Â = operator of an observable property
• [ ] is the commutator
• ${\displaystyle \langle \,\rangle }$  denotes the average
${\displaystyle {\frac {d}{dt}}{\hat {A}}(t)={\frac {i}{\hbar }}[{\hat {H}},{\hat {A}}(t)]+{\frac {\partial {\hat {A}}(t)}{\partial t}},}$
Time evolution in Heisenberg picture (Ehrenfest theorem)

of a particle.

${\displaystyle {\frac {d}{dt}}\langle {\hat {A}}\rangle ={\frac {1}{i\hbar }}\langle [{\hat {A}},{\hat {H}}]\rangle +\left\langle {\frac {\partial {\hat {A}}}{\partial t}}\right\rangle }$

For momentum and position;

${\displaystyle m{\frac {d}{dt}}\langle \mathbf {r} \rangle =\langle \mathbf {p} \rangle }$

${\displaystyle {\frac {d}{dt}}\langle \mathbf {p} \rangle =-\langle \nabla V\rangle }$

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 ${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x)=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)}$ {\displaystyle {\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 xn. ${\displaystyle E\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi +V\Psi }$ ${\displaystyle E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.}$ ${\displaystyle \Psi (x,t)=\psi (x)e^{-iEt/\hbar }\,.}$  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]${\displaystyle \|\psi \|^{2}=\int |\psi (x)|^{2}\,dx.\,}$ ${\displaystyle \Psi =e^{-iEt/\hbar }\psi (x_{1},x_{2}\cdots x_{N})}$  for non-interacting particles ${\displaystyle \Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (x_{n})\,,\quad V(x_{1},x_{2},\cdots x_{N})=\sum _{n=1}^{N}V(x_{n})\,.}$ Three dimensions {\displaystyle {\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). {\displaystyle {\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 = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is ${\displaystyle \nabla _{n}^{2}={\frac {\partial ^{2}}{{\partial x_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial y_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial z_{n}}^{2}}}}$ ${\displaystyle E\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi }$ ${\displaystyle E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi }$ ${\displaystyle \Psi =\psi (\mathbf {r} )e^{-iEt/\hbar }}$ ${\displaystyle \Psi =e^{-iEt/\hbar }\psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N})}$  for non-interacting particles ${\displaystyle \Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (\mathbf {r} _{n})\,,\quad V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})=\sum _{n=1}^{N}V(\mathbf {r} _{n})}$

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

Photoemission

Property/Effect Nomenclature Equation
Photoelectric equation
• Kmax = Maximum kinetic energy of ejected electron (J)
• h = Planck's constant
• f = frequency of incident photons (Hz = s−1)
• φ, Φ = Work function of the material the photons are incident on (J)
${\displaystyle K_{\mathrm {max} }=hf-\Phi \,\!}$
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:

${\displaystyle \phi =hf_{0}\,\!}$

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:

${\displaystyle p=hf/c=h/\lambda \,\!}$

Quantum uncertainty

Property or effect Nomenclature Equation
Heisenberg's uncertainty principles
• n = number of photons
• φ = wave phase
• [, ] = commutator
Position-momentum

${\displaystyle \sigma (x)\sigma (p)\geq {\frac {\hbar }{2}}\,\!}$

Energy-time ${\displaystyle \sigma (E)\sigma (t)\geq {\frac {\hbar }{2}}\,\!}$

Number-phase ${\displaystyle \sigma (n)\sigma (\phi )\geq {\frac {\hbar }{2}}\,\!}$

Dispersion of observable A = observables (eigenvalues of operator)

{\displaystyle {\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) ${\displaystyle \sigma (A)\sigma (B)\geq {\frac {1}{2}}\langle i[{\hat {A}},{\hat {B}}]\rangle }$
Probability Distributions
Property or effect Equation
Density of states ${\displaystyle N(E)=8{\sqrt {2}}\pi m^{3/2}E^{1/2}/h^{3}\,\!}$
Fermi–Dirac distribution (fermions) ${\displaystyle P(E_{i})={\frac {g(E_{i})}{e^{(E-\mu )/kT}+1}}}$

where

• 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) ${\displaystyle P(E_{i})={\frac {g(E_{i})}{e^{(E_{i}-\mu )/kT}-1}}}$

Angular momentum

Property or effect Nomenclature Equation
Angular momentum quantum numbers
• s = spin quantum number
• ms = spin magnetic quantum number
• = Azimuthal quantum number
• m = azimuthal magnetic quantum number
• j = total angular momentum quantum number
• mj = total angular momentum magnetic quantum number

Spin: {\displaystyle {\begin{aligned}&\Vert \mathbf {s} \Vert ={\sqrt {s\,(s+1)}}\,\hbar \\&m_{s}\in \{-s,-s+1\cdots s-1,s\}\\\end{aligned}}\,\!}

Orbital: {\displaystyle {\begin{aligned}&\ell \in \{0\cdots n-1\}\\&m_{\ell }\in \{-\ell ,-\ell +1\cdots \ell -1,\ell \}\\\end{aligned}}\,\!}

Total: {\displaystyle {\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:

${\displaystyle |\mathbf {S} |=\hbar {\sqrt {s(s+1)}}\,\!}$

Orbital magnitude: ${\displaystyle |\mathbf {L} |=\hbar {\sqrt {\ell (\ell +1)}}\,\!}$

Total magnitude: ${\displaystyle \mathbf {J} =\mathbf {L} +\mathbf {S} \,\!}$

${\displaystyle |\mathbf {J} |=\hbar {\sqrt {j(j+1)}}\,\!}$

Angular momentum components Spin:

${\displaystyle S_{z}=m_{s}\hbar \,\!}$

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

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
${\displaystyle {\boldsymbol {\mu }}_{\ell }=-e\mathbf {L} /2m_{e}=g_{\ell }{\frac {\mu _{B}}{\hbar }}\mathbf {L} \,\!}$

z-component: ${\displaystyle \mu _{\ell ,z}=-m_{\ell }\mu _{B}\,\!}$

spin magnetic dipole moment
${\displaystyle {\boldsymbol {\mu }}_{s}=-e\mathbf {S} /m_{e}=g_{s}{\frac {\mu _{B}}{\hbar }}\mathbf {S} \,\!}$

z-component: ${\displaystyle \mu _{s,z}=-eS_{z}/m_{e}=g_{s}eS_{z}/2m_{e}\,\!}$

dipole moment potential U = potential energy of dipole in field ${\displaystyle U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!}$

The Hydrogen atom

Property or effect Nomenclature Equation
Energy level
${\displaystyle E_{n}=-me^{4}/8\varepsilon _{0}^{2}h^{2}n^{2}=-13.61\,\mathrm {eV} /n^{2}}$
Spectrum λ = wavelength of emitted photon, during electronic transition from Ei to Ej ${\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}