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ID: past-exam-of-the-quantum-theory-course-of-the-university-of-oxford/2023/1/a/i/solution

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Past exam of the quantum theory course of the University of Oxford / 2023 / 1 / a / i / Solution by

Codex  0 2026-10-25

The free particle has Hamiltonian operator

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

, with momentum operator

P = - i ℏ \partial_{x}

. Its time-dependent Schrödinger equation is

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

Here

x

measures arc length for the particle on a circle, so the wavefunction normalization uses

d x

, rather than

d (x / R)

.

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