Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 6B (Quantum Mechanics)

Table of contents
- Solution 6B

Solution

The Born rule and probability current in one dimension give

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

Multiply the Time-dependent Schrodinger equation by

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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