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, also known as the reduced Planck constant or Dirac constant.
Quantity symbol/sDefining equationSI unitDimension
Wavefunctionψ, ΨTo solve from the Schrödinger equationvaries with situation and number of particles
Wavefunction probability densityρm−3−3
Wavefunction probability currentjNon-relativistic, no external field:
star * is complex conjugate		m−2⋅s−1−1 −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. Following are general mathematical results, used in calculations.
Property or effectNomenclatureEquation
Wavefunction for N particles in 3dr = sz =
In function notation:
in bra–ket notation:
for non-interacting particles:
Position-momentum Fourier transform
  • Φ = momentum–space wavefunction
  • Ψ = position–space wavefunction
General probability distributionVj = volume 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.
General normalization condition

Equations

Wave–particle duality and time evolution

Property or effectNomenclatureEquation
Planck–Einstein equation and de Broglie wavelength relationsP = is the four-momentum,K = is the four-wavevector,E = energy of particleω = 2πf is the angular frequency and frequency of the particleħ = h/2π are the Planck constantsc = speed of light
Schrödinger equationΨ = wavefunction of the systemĤ = Hamiltonian operator,E = energy eigenvalue of systemi is the imaginary unitt = time
General time-dependent case:
Time-independent case:
Heisenberg equation = operator of an observable property
Time evolution in Heisenberg picture m = mass,V = potential energy, r = position,p = momentum,
of a particle.
For momentum and position;

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
where the position of particle n is xn.
One dimension
One dimension
There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm or a slowly diverging norm :
for non-interacting particles
Three dimensions
where the position of the particle is r =.
where the position of particle n is r n =, and the Laplacian for particle n using the corresponding position coordinates is
Three dimensions
Three dimensions
for non-interacting particles

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
where the position of particle n is xn.
One dimension
One dimension
Three dimensions
Three dimensions
This last equation is in a very high dimension, so the solutions are not easy to visualize.
Three dimensions

Photoemission

Property/EffectNomenclatureEquation
Photoelectric equationKmax = Maximum kinetic energy of ejected electron h = Planck constantf = frequency of incident photons φ, Φ = Work function of the material the photons are incident on
Threshold frequency and Work functionφ, Φ = Work function of the material the photons are incident on f0, ν0 = Threshold frequency
Can only be found by experiment.
The De Broglie relations give the relation between them:
Photon momentump = momentum of photon f = frequency of photon λ = wavelength of photon

The De Broglie relations give:

Quantum uncertainty

Property or effectNomenclatureEquation
Heisenberg's uncertainty principlesn = number of photonsφ = wave phase
Position–momentum
Energy-time
Number-phase
Dispersion of observableA = observables
General uncertainty relationA, B = observables

Property or effectEquation
Density of states
Fermi–Dirac distribution
whereP = probability of energy Eig = degeneracy of energy Ei μ = chemical potential
Bose–Einstein distribution

Angular momentum

Property or effectNomenclatureEquation
Angular momentum quantum numberss = spin quantum numberms = spin magnetic quantum numberℓ = Azimuthal quantum numbermℓ = azimuthal magnetic quantum numberj = total angular momentum quantum numbermj = total angular momentum magnetic quantum number

Spin:
Orbital:
Total:
Angular momentum magnitudesangular momementa:S = Spin,L = orbital,J = total
Spin magnitude:
Orbital magnitude:
Total magnitude:
Angular momentum componentsSpin:
Orbital:

; Magnetic moments :
In what follows, B is an applied external magnetic field and the quantum numbers above are used.
Property or effectNomenclatureEquation
orbital magnetic dipole momente = electron chargeme = electron rest massL = electron orbital angular momentumg = orbital Landé g-factorμB = Bohr magneton

z-component:
spin magnetic dipole momentS = electron spin angular momentumgs = spin Landé g-factor

z-component:
dipole moment potentialU = potential energy of dipole in field

Hydrogen atom

Property or effectNomenclatureEquation
Energy levelEn = energy eigenvaluen = principal quantum numbere = electron chargeme = electron rest massε0 = permittivity of free spaceh = Planck constant
Spectrumλ = wavelength of emitted photon, during electronic transition from Ei to Ej