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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-1/a1/b/v/solution

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Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A1 / b / v / Solution by

Codex  0 2026-10-20

Project the first-order equation onto

Ψ_{j}^{(0)}

, with

j \neq = i

. By orthonormality and the Hermitian operator property,

⟨ Ψ_{j}^{(0)} ∣ (\hat{H}^{(0)} - E_{i}^{(0)}) ∣ Ψ_{i}^{(1)} ⟩ ⟨ Ψ_{j}^{(0)} ∣ (\hat{H}^{(1)} - E_{i}^{(1)}) ∣ Ψ_{i}^{(0)} ⟩ = (E_{j}^{(0)} - E_{i}^{(0)}) c_{ij}, = H_{ji} .

Consequently

c_{ij} = \frac{H _{ji}}{E _{i}^{(0)} - E _{j}^{(0)}} = \frac{H _{ij}^{*}}{E _{i}^{(0)} - E _{j}^{(0)}} (j \neq = i),

and

Ψ_{i}^{(1)} = j \neq = i \sum \frac{⟨ Ψ _{j}^{(0)} ∣ H ^ ^{(1)} ∣ Ψ _{i}^{(0)} ⟩}{E _{i}^{(0)} - E _{j}^{(0)}} Ψ_{j}^{(0)} .

The numerator order is important: the question defines

H_{ij}

with

i

on the bra, whereas projecting the correction onto state

j

gives

H_{ji}

. The two coincide only when the relevant matrix element is real.

Excluding an

i

component chooses

⟨ Ψ_{i}^{(0)} ∣ Ψ_{i}^{(1)} ⟩ = 0

, a permitted first-order phase and wavefunction normalization convention. A component parallel to

Ψ_{i}^{(0)}

is not determined by the projected equation. Nondegeneracy ensures nonzero denominators; a small energy gap can still make the perturbative expansion inaccurate unless

∣ λ H_{ji} ∣ ≪ ∣ E_{i}^{(0)} - E_{j}^{(0)} ∣

.

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