4D (Quantum Mechanics)

⟨ O ⟩_{ψ} = ⟨ ψ ∣ O ∣ ψ ⟩ .

Differentiating and retaining the possible explicit time dependence of the observable gives

\frac{d}{d t} ⟨ O ⟩_{ψ} = ⟨ \dot{ψ} ∣ O ∣ ψ ⟩ + ⟨ ψ \frac{\partial O}{\partial t} ψ ⟩ + ⟨ ψ ∣ O ∣ \dot{ψ} ⟩ .

i ℏ∣ \dot{ψ} ⟩ = H ∣ ψ ⟩, - i ℏ ⟨ \dot{ψ} ∣ = ⟨ ψ ∣ H,

where the Hamiltonian operator is Hermitian. Substitution yields

\frac{d}{d t} ⟨ O ⟩_{ψ} = \frac{i}{ℏ} ⟨ ψ ∣ H O ∣ ψ ⟩ - \frac{i}{ℏ} ⟨ ψ ∣ O H ∣ ψ ⟩ + ⟨ \frac{\partial O}{\partial t} ⟩_{ψ} = \frac{i}{ℏ} ⟨[H, O] ⟩_{ψ} + ⟨ \frac{\partial O}{\partial t} ⟩_{ψ} .

b

U (x)

H = \frac{p ^{2}}{2 m} + U (x) .

[x, p] = i ℏ

gives

[H, x] = \frac{1}{2 m} [p^{2}, x] = - \frac{i ℏ}{m} p .

O = x

therefore gives

m \frac{d}{d t} ⟨ x ⟩_{ψ} = ⟨ p ⟩_{ψ} .

Since

[p^{2}, p] = 0

and direct action on a wavefunction gives

[U (x), p] = i ℏ U^{'} (x)

\frac{d}{d t} ⟨ p ⟩_{ψ} = - ⟨ U^{'} (x) ⟩_{ψ} .

Finally

H

has no explicit time dependence and

[H, H] = 0

, so

\frac{d}{d t} ⟨ H ⟩_{ψ} = 0 .

The first two equations are the expectation-value counterparts of momentum equals mass times velocity and Newton's second law; the last is conservation of energy. They become the classical equations directly when the force varies negligibly across the wave packet, so that

⟨ U^{'} (x)⟩ ≃ U^{'} (⟨ x ⟩)

. These conclusions are summarized by the classical equations from Ehrenfest theorem.

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