bibkey: wikipedia2026stationarystate
authors: Wikipedia contributors
year: 2026
title: Stationary state
doi: null
url: https://en.wikipedia.org/wiki/Stationary_state
claim: An energy eigenstate evolves by an overall complex phase while all observable properties remain constant.
strata_touched:
  - D5/S3/Quantum/Dynamics/EnergyEigenstateStationarity
license: CC-BY-SA-4.0
triage: anchor

Stationary energy eigenstates

Verified locator

https://en.wikipedia.org/wiki/Stationary_state

The introduction gives H psi = E psi and the time evolution psi(t) = exp(-i E t / hbar) psi(0). The section on stationary-state properties explains that the overall complex phase cancels from observable probabilities. For the rank-one density matrix the same cancellation reads rho(t) = psi(t) psi(t)* = psi(0) psi(0)*.

The finite-dimensional formal statement uses hbar = 1, permits every real time, and derives the vector evolution from the convergent matrix exponential series. The zero-variance companion uses the density-state first and second moments.