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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2024/paper-1/a2/e/solution

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Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A2 / e / Solution by

Codex  0 2026-10-20

Write

\hat{N} = \overset{a}{^}^{†} \overset{a}{^}

. The creation and annihilation operators obey

[\hat{N}, \overset{a}{^}^{†}] = \overset{a}{^}^{†} [\overset{a}{^}, \overset{a}{^}^{†}] = \overset{a}{^}^{†}, [\hat{H}, \overset{a}{^}^{†}] = ℏ ω \overset{a}{^}^{†} .

(1)

Consequently,

\hat{H} \overset{a}{^}^{†} Ψ_{0} = (\overset{a}{^}^{†} \hat{H} + [\hat{H}, \overset{a}{^}^{†}]) Ψ_{0} = (E_{0} + ℏ ω) \overset{a}{^}^{†} Ψ_{0} = \frac{3}{2} ℏ ω \overset{a}{^}^{†} Ψ_{0} .

(2)

The resulting function is not zero: its squared norm is

∥ \overset{a}{^}^{†} Ψ_{0} ∥^{2} = ⟨ Ψ_{0} ∣ \overset{a}{^} \overset{a}{^}^{†} ∣ Ψ_{0} ⟩ = ⟨ \overset{a}{^}^{†} \overset{a}{^} + 1 ⟩ = 1.

(3)

Thus $\overset{a}{^}^{†} Ψ_{0}$ is the normalized first excited state, up to phase. More generally the creation operator raises the energy by one quantum

ℏ ω

\overset{a}{^}^{†} ∣ n ⟩ = n + 1 ∣ n + 1 ⟩

. The factor is required because applying it to a higher state does not preserve normalization.

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