Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 1 / b / i

Table of contents
- Solution i

Solution

The particle on a circle has the complete orthonormal basis of energy eigenstates

{∣ n ⟩ : n \in Z}

. Its spectral representation gives

K (θ_{f}, T; θ_{i}, 0) = n \in Z \sum ⟨ θ_{f} ∣ n ⟩ e^{- i E_{n} T /ℏ} ⟨ n ∣ θ_{i} ⟩ = \frac{1}{2 π} n \in Z \sum exp [in (θ_{f} - θ_{i}) - \frac{i n ^{2} ℏ T}{2 m R ^{2}}] .

n

is the quantized angular momentum in units of

ℏ

.

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