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 (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 m−3 [L]−3 Wavefunction probability current j Non-relativistic, no external field:

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:

in bra–ket notation:

for non-interacting particles: Position-momentum Fourier transform (1 particle in 3d) Φ = momentum–space wavefunction Ψ = position–space wavefunctionGeneral 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. General normalization condition

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 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:

Time-independent case: Heisenberg equation  = operator of an observable property [ ] is the commutator denotes the average Time evolution in Heisenberg picture (Ehrenfest theorem) 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.

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):

for non-interacting particles Three dimensions

where the position of the particle is r = (x, y, z).

where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is

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. Three dimensions

This last equation is in a very high dimension, so the solutions are not easy to visualize.

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) 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: 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:

Quantum uncertainty

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

Energy-time

Number-phaseDispersion of observableA = observables (eigenvalues of operator) General uncertainty relation A, B = observables (eigenvalues of operator)

+Probability Distributions Property or effect Equation Density of states Fermi–Dirac distribution (fermions) 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)

Angular momentum

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 numberSpin:

Orbital:

Total: Angular momentum magnitudes angular momementa: S = Spin, L = orbital, J = total Spin magnitude:

Orbital magnitude:

Total magnitude: Angular momentum components Spin:

Orbital:

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 magnetonz-component: spin magnetic dipole moment S = electron spin angular momentum gs = spin Landé g-factor

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

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 Spectrum λ = wavelength of emitted photon, during electronic transition from Ei to Ej

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

Footnotes

Sources

Further reading