Energy eigenstates

A stationary state is a purequantum state with all observables independent of time. It is an eigenvector of the Hamiltonian. This corresponds to a state with a single definite energy (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket. It is very similar to the concept of atomic orbital and molecular orbital in chemistry, with some slight differences explained below.

A stationary state is called stationary because the system remains in the same state as time elapses, in every observable way. For a single-particle Hamiltonian, this means that the particle has a constant probability distribution for its position, its velocity, its spin, etc. (This is true assuming the particle's environment is also static, i.e. the Hamiltonian is unchanging in time.) The wavefunction itself is not stationary: It continually changes its overall complex phase factor, so as to form a standing wave. The oscillation frequency of the standing wave, times Planck's constant, is the energy of the state according to the Planck–Einstein relation.

Stationary states are quantum states that are solutions to the time-independent Schrödinger equation:

where

This is an eigenvalue equation: ${\displaystyle {\hat {H}}}$ is a linear operator on a vector space, ${\displaystyle |\Psi \rangle }$ is an eigenvector of ${\displaystyle {\hat {H}}}$, and ${\displaystyle E_{\Psi }}$ is its eigenvalue.

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