A stationary state is a quantum state with all observables independent of time. It is an eigenvector of the energy operator (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, multiplied by the Planck 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: $${\displaystyle {\hat {H}}|\Psi \rangle =E_{\Psi }|\Psi \rangle ,}$$ 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.

If a stationary state ${\displaystyle |\Psi \rangle }$ is plugged into the time-dependent Schrödinger equation, the result is $${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\Psi \rangle =E_{\Psi }|\Psi \rangle .}$$

Assuming that ${\displaystyle {\hat {H}}}$ is time-independent (unchanging in time), this equation holds for any time t. Therefore, this is a differential equation describing how ${\displaystyle |\Psi \rangle }$ varies in time. Its solution is $${\displaystyle |\Psi (t)\rangle =e^{-iE_{\Psi }t/\hbar }|\Psi (0)\rangle .}$$

Therefore, a stationary state is a standing wave that oscillates with an overall complex phase factor, and its oscillation angular frequency is equal to its energy divided by ${\displaystyle \hbar }$.

As shown above, a stationary state is not mathematically constant: $${\displaystyle |\Psi (t)\rangle =e^{-iE_{\Psi }t/\hbar }|\Psi (0)\rangle .}$$