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 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

Equations

Wave–particle duality and time evolution

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 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):[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 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,[2] 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'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 uncertainty

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 Nomenclature Equation
Density of states
Fermi–Dirac distribution (fermions)
  • 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
  • = 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 atom

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

See also

Footnotes

  1. Feynman, R.P.; Leighton, R.B.; Sand, M. (1964). "Operators". The Feynman Lectures on Physics. 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.

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.
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