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Time dependence

For systems in time-independent potentials, the wave function can always be written as a function of the degrees of freedom multiplied by a time-dependent phase factor, the form of which is given by the Schrödinger equation. For N particles, considering their positions only and suppressing other degrees of freedom, Ψ (r 1, r 2, …, r N, t) = e − i E t / ℏ ψ (r 1, r 2, …, r N), {\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2},\ldots,\mathbf {r} _{N},t)=e^{-iEt/\hbar }\,\psi (\mathbf {r} _{1},\mathbf {r} _{2},\ldots,\mathbf {r} _{N})\,,} where E is the energy eigenvalue of the system corresponding to the eigenstate Ψ. Wave functions of this form are called stationary states.

The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states. For any quantum state | Ψ ⟩ and operator O, in the Schrödinger picture | Ψ(t) ⟩ changes with time according to the Schrödinger equation while O is constant. In the Heisenberg picture it is the other way round, | Ψ ⟩ is constant while O (t) evolves with time according to the Heisenberg equation of motion. The Dirac (or interaction) picture is intermediate, time dependence is places in both operators and states which evolve according to equations of motion. It is useful primarily in computing S-matrix elements.