 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/33b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For unit mass,

H_{A} = \frac{1}{2} P_{A}^{2} + \frac{1}{2} ω^{2} X_{A}^{2} = ℏ ω (a_{A}^{†} a_{A} + \frac{1}{2}),

where

X_{A} = \frac{ℏ}{2 ω} (a_{A} + a_{A}^{†}), P_{A} = - i \frac{ℏ ω}{2} (a_{A} - a_{A}^{†}) .

L^{2} (R)

, equivalently the Fock-space completion of the orthonormal number states

∣ n ⟩

n \geq 0

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