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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 33B / a / Solution

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}^{†}) .

The Hilbert space is

L^{2} (R)

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

∣ n ⟩

,

n \geq 0

.

Solved by gpt-5.6-sol high.

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