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 Units	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_{z i}. 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 = (r1, r2... rN) sz = (sz 1, sz 2, ..., sz 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	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.	$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int _{V_{N}}\cdots \int _{V_{2}}\int _{V_{1}}\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,\!$
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	 = 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 xn.
	$E\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi +V\Psi$	$E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi (x,t)=\psi (x)e^{-iEt/\hbar }\,.$ 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): $\|\psi \|^{2}=\int |\psi (x)|^{2}\,dx.\,$	$\Psi =e^{-iEt/\hbar }\psi (x_{1},x_{2}\cdots x_{N})$ 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 = (xn, yn, zn), 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 $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (\mathbf {r} _{n})\,,\quad V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})=\sum _{n=1}^{N}V(\mathbf {r} _{n})$

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 xn.
	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi +V\Psi$	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi =\Psi (x,t)$	$\Psi =\Psi (x_{1},x_{2}\cdots x_{N},t)$
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, 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

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)	$K_{\mathrm {max} }=hf-\Phi \,\!$
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: $p=hf/c=h/\lambda \,\!$

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(Ei) = probability of energy Ei g(Ei) = degeneracy of energy Ei (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

Main articles: angular momentum operator and quantum number

Property or effect	Nomenclature	Equation
Angular momentum quantum numbers	s = spin quantum number ms = spin magnetic quantum number ℓ = Azimuthal quantum number mℓ = azimuthal magnetic quantum number j = total angular momentum quantum number mj = 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 me = electron rest mass L = electron orbital angular momentum gℓ = orbital Landé g-factor μB = 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 gs = 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	$U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!$

The Hydrogen atom

Main article: Hydrogen atom

Property or effect	Nomenclature	Equation
Energy level	En = energy eigenvalue n = principal quantum number e = electron charge me = electron rest mass ε0 = permittivity of free space h = Planck's 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 Ei to Ej	${\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}<n_{i}\,\!$

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

Sources

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

Further reading

• L. H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W. B. Saunders and Co. ISBN 0-7216-4247-0.
• J. B. Marion; W. F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2.
• A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1.
• H. D. Young; R. A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 978-0-321-50130-1.