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ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2021/paper-1/section-b/b7/a/solution

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Past exam of the physics A course of the University of Cambridge / 2021 / Paper 1 / Section B / B7 / a / Solution by

Codex  0 2026-10-18

Write

a = a_{-}

and

a^{†} = a_{+}

. Inverting the annihilation operator and creation operator definitions gives

x = \frac{ℏ}{2 mω} (a + a^{†}), p = i \frac{m ℏ ω}{2} (a^{†} - a) .

The ladder operators take

∣ n ⟩

to orthogonal quantum harmonic oscillator energy eigenstates

∣ n - 1 ⟩

and

∣ n + 1 ⟩

. Thus

⟨ n ∣ a ∣ n ⟩ = ⟨ n ∣ a^{†} ∣ n ⟩ = 0

, and

⟨ x ⟩_{n} = 0, ⟨ p ⟩_{n} = 0.

For

n = 0

, the lowering term is simply zero. The conclusion concerns an individual energy eigenstate; interference between different eigenstates can produce nonzero expectation values.

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