List of equations in quantum mechanics

This article summarizes equations in the theory of quantum mechanics.

WavefunctionsEdit

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   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 wavefunction
 
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.
 
General normalization condition  

EquationsEdit

Wave–particle duality and time evolutionEdit

Property or effect Nomenclature Equation
Planck–Einstein equation and de Broglie wavelength relations
 
Schrödinger equation
General time-dependent case:

 

Time-independent case:  

Heisenberg equation
  • Â = operator of an observable property
  • [ ] is the commutator
  •   denotes the average
 
Time evolution in Heisenberg picture (Ehrenfest theorem)

of a particle.

 

For momentum and position;

 

 

Non-relativistic time-independent Schrödinger equationEdit

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):[1] 

 

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 equationEdit

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,[2] so the solutions are not easy to visualize.

   

PhotoemissionEdit

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)
 
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 uncertaintyEdit

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

 

Energy-time  

Number-phase  

Dispersion of observable A = 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 momentumEdit

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:  

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
 

z-component:  

spin magnetic dipole moment
 

z-component:  

dipole moment potential U = potential energy of dipole in field  

The Hydrogen atomEdit

Property or effect Nomenclature Equation
Energy level
 
Spectrum λ = wavelength of emitted photon, during electronic transition from Ei to Ej  

See alsoEdit

FootnotesEdit

  1. ^ Feynman, R.P.; Leighton, R.B.; Sand, M. (1964). "Operators". The Feynman Lectures on Physics. Vol. 3. Addison-Wesley. pp. 20–7. ISBN 0-201-02115-3.
  2. ^ Shankar, R. (1994). Principles of Quantum Mechanics. Kluwer Academic/Plenum Publishers. p. 141. ISBN 978-0-306-44790-7.

SourcesEdit

Further readingEdit