Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 6B / b

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- Solution b

Solution

The probability density and probability current are

ρ = ∣ ψ ∣^{2}, J = \frac{ℏ}{2 mi} (ψ^{*} \frac{\partial ψ}{\partial x} - ψ \frac{\partial ψ ^{*}}{\partial x}) .

For a stationary state,

ψ (x, t) = χ (x) e^{- i Et /ℏ}

, so

ρ = ∣ χ ∣^{2}

is independent of time. The continuity equation then gives

\partial J / \partial x = 0

. Hence

J is independent of x .

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