List of equations in quantum mechanics explained

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, 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{align} j&= -i\hbar 2m \left(\Psi*\nabla\Psi-\Psi\nabla\Psi*\right)\\ &= \hbar m \operatorname{Im}\left(\Psi*\nabla\Psi\right)=\operatorname{Re}\left(\Psi* \hbar im \nabla\Psi\right) \end{align} 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(r, sz, t\right) in bra–ket notation:	\Psi\rangle = \sum_ \sum_\cdots\sum_\int_\int_\cdots\int_ \mathrm\mathbf_1\mathrm\mathbf_2\cdots\mathrm\mathbf_N \Psi	\mathbf, \mathbf\rangle for non-interacting particles: \Psi= N\Psi \prod n=1 \left(rn,szn,t\right)
Position-momentum Fourier transform (1 particle in 3d)	Φ = momentum–space wavefunction Ψ = position–space wavefunction	\begin{align}\Phi(p,sz,t)&= 1 \sqrt{2\pi\hbar 3}\int\limitsallspacee-ip ⋅ r/\hbar\Psi(r, 3r s z,t)d \  &\upharpoonleft\downharpoonright\\ \Psi(r,sz,t)&= 1 \sqrt{2\pi\hbar 3}\int\limitsallspacee+ip ⋅ r/\hbar 3p \Phi(p,s n \\ \end{align}
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 szN … \sum sz2 \sum sz1 \int VN … \int V2 \int V1 \left	\Psi \right	^2\mathrm^3\mathbf_1\mathrm^3\mathbf_2\cdots\mathrm^3\mathbf_N\,\!
General normalization condition		P= \sum szN … \sum sz2 \sum sz1 \int\limitsallspace … \int\limitsallspace \int\limitsallspace\left	\Psi \right	^2\mathrm^3\mathbf_1\mathrm^3\mathbf_2\cdots\mathrm^3\mathbf_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	P=(E/c,p)=\hbar(\omega/c,k)=\hbarK
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 \partial \partialt \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	d \hat{A}(t)= dt i [\hat{H},\hat{A}(t)]+ \hbar \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.	d dt \langle\hat{A}\rangle= 1 i\hbar \langle[\hat{A},\hat{H}]\rangle+\left\langle \partial\hat{A }\right\rangle For momentum and position; m d dt \langler\rangle=\langlep\rangle d dt \langlep\rangle=-\langle\nablaV\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}= \hat{p 2}{2m} +V(x)=- \hbar2 2m d2 dx2 +V(x)	\begin{align} \hat{H}&= N \sum n=1 \hat{p n 2}{2m n} +V(x1,x2, … xN)\  &=- \hbar2 2 N \sum n=1 1 mn \partial2 \partial 2 x n +V(x1,x2, … xN)\end{align} where the position of particle n is xn.
	E\Psi=- \hbar2 2m d2 dx2 \Psi+V\Psi	E\Psi=- \hbar2 2 N \sum n=1 1 mn \partial2 \partial 2 x n \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):[1] \	\psi \	^2 = \int	\psi(x)	^2\, dx.\,	\Psi=e-iEt/\hbar\psi(x1,x2 … xN) for non-interacting particles \Psi=e-i{E }\prod_^N\psi(x_n) \,, \quad V(x_1,x_2,\cdots x_N) = \sum_^N V(x_n) \, .
Three dimensions	\begin{align}\hat{H}&= \hat{p ⋅ \hat{p }} + V(\mathbf) \\& = -\frac\nabla^2 + V(\mathbf) \end where the position of the particle is r = (x, y, z).	\begin{align}\hat{H}&= N \sum n=1 \hat{p n ⋅ \hat{p }_n} + V(\mathbf_1,\mathbf_2,\cdots\mathbf_N) \\& = -\frac\sum_^\frac\nabla_n^2 + V(\mathbf_1,\mathbf_2,\cdots\mathbf_N) \end where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is 2= \partial2 {\partialxn \nabla n 2}+ \partial2 {\partialyn 2}+ \partial2 {\partialzn 2}
	E\Psi=- \hbar2 2m \nabla2\Psi+V\Psi	E\Psi=- \hbar2 2 N \sum n=1 1 mn 2\Psi \nabla n +V\Psi
	\Psi=\psi(r)e-iEt/\hbar	\Psi=e-iEt/\hbar\psi(r1,r2 … rN) for non-interacting particles \Psi=e-i{E }\prod_^N\psi(\mathbf_n) \,, \quad V(\mathbf_1,\mathbf_2,\cdots \mathbf_N) = \sum_^N V(\mathbf_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}= \hat{p 2}{2m} +V(x,t)=- \hbar2 2m \partial2 \partialx2 +V(x,t)	\begin{align} \hat{H}&= N \sum n=1 \hat{p n 2}{2m n} +V(x1,x2, … xN,t)\\ &=- \hbar2 2 N \sum n=1 1 mn \partial2 \partial 2 x n +V(x1,x2, … xN,t) \end{align} where the position of particle n is xn.
	i\hbar \partial \partialt \Psi=- \hbar2 2m \partial2 \partialx2 \Psi+V\Psi	i\hbar \partial \partialt \Psi=- \hbar2 2 N \sum n=1 1 mn \partial2 \partial 2 x n \Psi+V\Psi.
	\Psi=\Psi(x,t)	\Psi=\Psi(x1,x2 … xN,t)
Three dimensions	\begin{align}\hat{H}&= \hat{p ⋅ \hat{p }} + V(\mathbf,t) \\& = -\frac\nabla^2 + V(\mathbf,t) \\\end	\begin{align}\hat{H}&= N \sum n=1 \hat{p n ⋅ \hat{p }_n} + V(\mathbf_1,\mathbf_2,\cdots\mathbf_N,t) \\& = -\frac\sum_^\frac\nabla_n^2 + V(\mathbf_1,\mathbf_2,\cdots\mathbf_N,t) \end
	i\hbar \partial \partialt \Psi=- \hbar2 2m \nabla2\Psi+V\Psi	i\hbar \partial \partialt \Psi=- \hbar2 2 N \sum n=1 1 mn 2\Psi \nabla n +V\Psi This last equation is in a very high dimension,[2] so the solutions are not easy to visualize.
	\Psi=\Psi(r,t)	\Psi=\Psi(r1,r2, … rN,t)

Photoemission

Property/Effect	Nomenclature	Equation
Photoelectric equation	Kmax = 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)	Kmax=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=hf0
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/λ

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)\ge \hbar 2 Energy-time \sigma(E)\sigma(t)\ge \hbar 2 \	Number-phase \sigma(n)\sigma(\phi)\ge \hbar 2
Dispersion of observable	A = observables (eigenvalues of operator)	\begin{align} \sigma(A)2&=\langle(A-\langleA\rangle)2\rangle\\ &=\langleA2\rangle-\langleA\rangle2 \end{align}
General uncertainty relation	A, B = observables (eigenvalues of operator)	\sigma(A)\sigma(B)\geq 1 2 \langlei[\hat{A},\hat{B}]\rangle

Property or effect ! scope="col" width="10"

Equation
Density of states	N(E)=8\sqrt{2}\pim3/2E1/2/h3
Fermi–Dirac distribution (fermions)	P(Ei)= g(Ei) 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(Ei)= g(Ei) (Ei-\mu)/kT e -1

Angular momentum

See main article: 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{align}&\Verts\Vert=\sqrt{s(s+1)}\hbar\\ &ms\in\{-s,-s+1 … s-1,s\}\\ \end{align} Orbital: \begin{align}&\ell\in\{0 … n-1\}\\ &m\ell\in\{-\ell,-\ell+1 … \ell-1,\ell\}\\ \end{align}\	Total: \begin{align}&j=\ell+s\\ &mj\in\{|\ell-s|,|\ell-s|+1 … |\ell+s|-1,|\ell+s|\}\\ \end{align}
Angular momentum magnitudes	angular momementa: S = Spin, L = orbital, J = total	Spin magnitude:	\mathbf	= \hbar\sqrt\,\! Orbital magnitude:	\mathbf	= \hbar\sqrt\,\! Total magnitude: J=L+S\		\mathbf	= \hbar\sqrt\,\!
Angular momentum components		Spin: Sz=ms\hbar Orbital: Lz=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 g = orbital Landé g-factor μB = Bohr magneton	\boldsymbol{\mu}\ell=-eL/2me=g\ell \muB \hbar L z-component: \mu\ell,z=-m\ell\muB\
spin magnetic dipole moment	S = electron spin angular momentum gs = spin Landé g-factor	\boldsymbol{\mu}s=-eS/me=gs \muB \hbar S z-component: \mus,z=-eSz/me=gseSz/2me\
dipole moment potential	U = potential energy of dipole in field	U=-\boldsymbol{\mu} ⋅ B=-\muzB

Hydrogen atom

See 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 constant	En=-me4/ 2 8\varepsilon 0 h2n2=-13.61eV/n2
Spectrum	λ = wavelength of emitted photon, during electronic transition from Ei to Ej	1 λ =R\left( 1 2 n j - 1 2 n i \right),nj<ni

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

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Further reading

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

Notes and References

1. Book: Feynman . R.P. . Leighton . R.B. . Sand . M. . 1964 . Operators . . 3 . 20–7 . . 0-201-02115-3.
2. Book: Shankar, R. . 1994 . Principles of Quantum Mechanics . limited . 141 . . 978-0-306-44790-7.