 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-4/6b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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

Ψ^{*}

, subtract its complex conjugate multiplied by

Ψ

, and rearrange to obtain the continuity equation

ρ_{t} + j_{x} = 0

For the stated superposition, the interference terms in

j

are real and cancel from its imaginary part. Hence

j = \frac{ℏ k}{m} (∣ A ∣^{2} - ∣ B ∣^{2}),

independent of

x

and

t

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