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It is a stationary state: its global phase changes, but its probability density and the distributions of time-independent observables do not. Its energy is sharp. Any quantum superposition entirely within one degenerate energy eigenspace also acquires one common global phase and is stationary. Stationarity does not require zero probability current; for instance,

u_{n} (ϕ) = e^{in ϕ} / 2 π

on the ring has a constant density and persistent circulation when

n \neq = 0

An arbitrary normalized state can instead contain different energy eigenvalues:

∣ ψ (t)⟩ = \sum_{n} c_{n} e^{- i E_{n} (t - t_{0}) /ℏ} ∣ n ⟩, \sum_{n} ∣ c_{n} ∣^{2} = 1.

E_{n}

with fixed probability

∣ c_{n} ∣^{2}

(summed over degeneracies), but relative quantum phases change with time. Consequently

∣ ψ (r, t) ∣^{2} = \sum_{m, n} c_{m}^{*} c_{n} ϕ_{m}^{*} (r) ϕ_{n} (r) e^{i (E_{m} - E_{n}) (t - t_{0}) /ℏ} .

(E_{m} - E_{n}) /ℏ

. For two populated energy levels, this supplies a characteristic period

2 π ℏ/∣ E_{m} - E_{n} ∣

when the interference term is nonzero. Thus fixed energy populations do not imply a stationary spatial distribution.

For a normalized state and an observable

\hat{A} (t)

⟨ A ⟩_{t} = ⟨ ψ (t) ∣ \hat{A} (t) ∣ ψ (t)⟩ .

This is the statistical mean over repeated preparations and measurements. In an eigenbasis of energy eigenstates for a time-independent Hamiltonian and a time-independent

\hat{A}

⟨ A ⟩_{t} = \sum_{m, n} c_{m}^{*} c_{n} A_{mn} e^{i (E_{m} - E_{n}) (t - t_{0}) /ℏ} .

Only matrix elements connecting different energies can generate these oscillations. If

[\hat{A}, \hat{H}] = 0

, its expectation value is constant for every initial state, provided it has no explicit time dependence.

Differentiating the expectation value and substituting the Schrödinger equation for both the bra and ket proves the general Ehrenfest theorem:

\frac{d}{d t} ⟨ A ⟩ = ⟨ \dot{ψ} ∣ \hat{A} ∣ ψ ⟩ + ⟨ \frac{\partial A ^}{\partial t} ⟩ + ⟨ ψ ∣ \hat{A} ∣ \dot{ψ} ⟩ = \frac{i}{ℏ} ⟨ \hat{H} \hat{A} - \hat{A} \hat{H} ⟩ + ⟨ \frac{\partial A ^}{\partial t} ⟩ .

Equivalently,

\frac{d}{d t} ⟨ A ⟩ = \frac{1}{i ℏ} ⟨[\hat{A}, \hat{H}]⟩ + ⟨ \frac{\partial A ^}{\partial t} ⟩ .

These manipulations require the state to lie in the relevant operator domains. Taking

\hat{A} = \hat{H}

gives

d ⟨ H ⟩ / d t = ⟨ \partial_{t} \hat{H} ⟩

, so energy is conserved under a time-independent Hamiltonian.

For

\hat{H} = \overset{p}{^}^{2} / (2 m) + V (\overset{x}{^})

[\overset{x}{^}, \hat{H}] = \frac{i ℏ}{m} \overset{p}{^}, [\overset{p}{^}, \hat{H}] = - i ℏ V^{'} (\overset{x}{^}) .

Hence the particularly useful position-and-momentum form of Ehrenfest theorem is

\frac{d}{d t} ⟨ x ⟩ = \frac{⟨ p ⟩}{m}, \frac{d}{d t} ⟨ p ⟩ = - ⟨ V^{'} (x)⟩ .

It resembles Newton's second law, but the mean force generally differs from the force at the mean position. If a narrow wave packet samples a region where the force varies little,

⟨ V^{'} (x)⟩ ≃ V^{'} (⟨ x ⟩)

gives approximate classical motion. For a quadratic potential energy this replacement is exact. In particular the quantum harmonic oscillator obeys

\frac{d ^{2}}{d t ^{2}} ⟨ x ⟩ = - ω^{2} ⟨ x ⟩, ⟨ x ⟩_{t} = ⟨ x ⟩_{t_{0}} cos ω τ + \frac{⟨ p ⟩ _{t_{0}}}{mω} sin ω τ, τ = t - t_{0} .

The centroid follows the classical harmonic oscillator, although this does not specify the full evolving wavefunction or its width.

The Heisenberg picture expresses the same physics by assigning evolution to observables instead:

\hat{A}_{H} (t) = U^{†} (t, t_{0}) \hat{A}_{S} (t) U (t, t_{0}), ∣ ψ_{H} ⟩ = ∣ ψ (t_{0})⟩ .

d \hat{A}_{H} / d t = (i /ℏ) [\hat{H}_{H}, \hat{A}_{H}] + U^{†} (\partial_{t} \hat{A}_{S}) U

. Thus state propagation, evolving expectation values, and the classical correspondence in Ehrenfest theorem are mutually consistent descriptions of the same unitary time evolution.