Past exam of the physics A course of the University of Cambridge / 2018 / Paper 1 / Section B / B6 / a

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- Solution a

Solution

Let

ρ = ψ^{*} ψ

be the probability density. For a real potential energy

V (x)

, the Schrödinger equation and its complex conjugate are

i ℏ \partial_{t} ψ = - \frac{ℏ ^{2}}{2 m} \partial_{x}^{2} ψ + V ψ, - i ℏ \partial_{t} ψ^{*} = - \frac{ℏ ^{2}}{2 m} \partial_{x}^{2} ψ^{*} + V ψ^{*} .

Differentiate

ρ

and substitute both equations. The potential terms cancel, leaving

\partial_{t} ρ = \frac{i ℏ}{2 m} (ψ^{*} \partial_{x}^{2} ψ - ψ \partial_{x}^{2} ψ^{*}) = \frac{i ℏ}{2 m} \partial_{x} (ψ^{*} \partial_{x} ψ - ψ \partial_{x} ψ^{*}) .

Comparison with the probability continuity equation

\partial_{t} ρ + \partial_{x} J = 0

identifies

J = \frac{i ℏ}{2 m} [(\partial_{x} ψ^{*}) ψ - ψ^{*} (\partial_{x} ψ)] = \frac{ℏ}{m} Im (ψ^{*} \partial_{x} ψ) .

For a plane wave of amplitude

A

and positive wavenumber

k

,

J = ℏ k ∣ A ∣^{2} / m

, the density times the velocity. Integrating the continuity equation over an interval makes its probability change equal to the incoming current minus the outgoing current.

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