Substitution of the stated phase into

i ℏ ψ_{t} = - \frac{ℏ ^{2}}{2 m} ψ_{xx} + m gx ψ

cancels the potential and scalar phase terms, leaving

i ℏ Φ_{t} = - \frac{ℏ ^{2}}{2 m} Φ_{xx} + i ℏ g t Φ_{x} .

ii

Solution

With

X = x + g t^{2} /2

and

T = t

, the chain rule gives

Φ_{t} = Θ_{T} + g t Θ_{X}

and

Φ_{x} = Θ_{X}

. The drift terms cancel, so

i ℏ Θ_{T} = - \frac{ℏ ^{2}}{2 m} Θ_{XX} .

Applying Ehrenfest's theorem with

[X, P] = i ℏ

and

[P, P^{2}] = 0

gives

\frac{d}{d T} ⟨ X ⟩ = \frac{1}{m} ⟨ P ⟩, \frac{d}{d T} ⟨ P ⟩ = 0 .

The phase has unit modulus and

X = x + g t^{2} /2

, hence

⟨ X ⟩_{Θ} = ⟨ x ⟩_{ψ} + \frac{1}{2} g t^{2} .

⟨ X ⟩ = a + v t

. Therefore

⟨ x ⟩_{ψ} = a + v t - \frac{1}{2} g t^{2} .

The wavepacket centre obeys the classical constant-acceleration trajectory, an instance of the correspondence principle.

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