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List of equations in quantum mechanics

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This article summarizes equations in the theory of quantum mechanics.

Wavefunctions

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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.

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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.

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Equations

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Wave–particle duality and time evolution

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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

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Quantum uncertainty

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Angular momentum

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Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

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Hydrogen atom

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Footnotes

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Further reading

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