Energy eigenstate ( $H ϕ = Eϕ$ )

An energy eigenstate is a quantum state represented by an eigenvector of the Quantum Hamiltonian. For a time-independent Quantum Hamiltonian, a normalized eigenfunction

ϕ

with energy eigenvalue

E

evolves according to the Schrödinger equation as

ψ (x, t) = ϕ (x) e^{- i Et /ℏ}, ∣ ψ (x, t) ∣^{2} = ∣ ϕ (x) ∣^{2} .

Here

ℏ = h / (2 π)

is the reduced Planck constant. The changing factor is a global phase, so this is a stationary state: its position probability density is constant. For example, the second infinite square well mode has two density lobes separated by a node of a standing wave at

x = L /2

; both lobes stay fixed while its quantum phase rotates. An ideal energy measurement returns

E

with probability one, but a position measurement still has a distribution. Stationary density does not describe a particle sitting at the peak of a lobe. Distinct energy eigenvalues accumulate different quantum phases, making their quantum superposition behave differently.

Table of contents
- Energy eigenfunction Energy eigenstate
- Energy eigenvector Energy eigenstate

Energy eigenfunction ( $\hat{H} ψ = E ψ$ )

A nonzero position-space wavefunction representing an energy eigenstate. For a time-independent Quantum Hamiltonian, its time evolution is a global phase, so its position probability density is stationary.

Energy eigenvector

A nonzero eigenvector of the Quantum Hamiltonian.

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