 Solution

ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2014/1/a/solution

m

V

, is

i ℏ \frac{\partial Ψ}{\partial t} = H Ψ, H = - \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x) .

Ψ (x, t) = ψ (x) e^{- i Et /ℏ}, - \frac{ℏ ^{2}}{2 m} ψ^{''} + V ψ = E ψ .

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

Ψ

, from the original equation multiplied by

Ψ^{*}

cancels the real potential energy terms and gives the continuity equation

\partial_{t} ρ + \partial_{x} j = 0

. Thus

j

measures signed probability flow in the positive

x

direction.

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