 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ib/paper-4/6b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

For the wavefunction of a particle of mass

m

ρ = Ψ^{*} Ψ, j = \frac{ℏ}{2 mi} (Ψ^{*} Ψ_{x} - Ψ Ψ_{x}^{*}) = \frac{ℏ}{m} Im (Ψ^{*} Ψ_{x}) .

For a real potential

V

i ℏ Ψ_{t} = - \frac{ℏ ^{2}}{2 m} Ψ_{xx} + V Ψ, - i ℏ Ψ_{t}^{*} = - \frac{ℏ ^{2}}{2 m} Ψ_{xx}^{*} + V Ψ^{*} .

Multiplying by

Ψ^{*}

and

Ψ

respectively yields

ρ_{t} = \frac{i ℏ}{2 m} (Ψ^{*} Ψ_{xx} - Ψ Ψ_{xx}^{*}) = - j_{x} .

The potential terms cancel because

V

is real. This is the continuity equation, expressing local conservation of probability.

For the given superposition the common time phase cancels in

Ψ^{*} Ψ_{x}

, and

Ψ^{*} Ψ_{x} = ik (1 - ∣ R ∣^{2}) + ik (R^{*} e^{2 ik x} - R e^{- 2 ik x}) .

The expression in the second parentheses is purely imaginary, so the second term is real and contributes nothing to the probability current. Thus

j = \frac{ℏ k}{m} (1 - ∣ R ∣^{2}) .

It is the incoming probability current minus the reflected probability current.

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