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ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2019/paper-1/section-b/b9/a/solution

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Past exam of the physics A course of the University of Cambridge / 2019 / Paper 1 / Section B / B9 / a / Solution by

Codex  0 2026-10-18

Let

c = mω / (2ℏ)

and

d = 1/ 2 m ℏ ω

, so

a = c x + i d p

and

a^{†} = c x - i d p

. The canonical commutation relation

[x, p] = i ℏ

gives

a^{†} a = c^{2} x^{2} + d^{2} p^{2} + i c d [x, p] = \frac{mω x ^{2}}{2ℏ} + \frac{p ^{2}}{2 m ℏ ω} - \frac{1}{2} .

Hence

H = ℏ ω (a^{†} a + 1/2)

. Similarly,

[a, a^{†}] = - 2 i c d [x, p] = 1 .

Using the product rule for a commutator,

[A B, C] = A [B, C] + [A, C] B

, gives

[H, a] = - ℏ ωa, [H, a^{†}] = ℏ ω a^{†} .

Thus

a^{†} a

is the number operator, while

a

and

a^{†}

are the annihilation operator and creation operator.

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