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ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2021/1/a/i/solution

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Past exam of the quantum theory course of the University of Oxford / 2021 / 1 / a / i / Solution by

Codex  0 2026-10-25

For a free particle, the Hamiltonian operator is

H = P^{2} / (2 m)

, where

P = - i ℏ d / d x

is the momentum operator. The time-independent Schrödinger equation is therefore

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} = E ψ, ψ (x + 2 π) = ψ (x) .

The full stationary state is

Ψ (x, t) = e^{- i Et /ℏ} ψ (x)

. The periodic boundary condition identifies

x

and

x + 2 π

, giving a particle on a circle with wavefunction normalization over one period.

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