Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A1 / b / iii

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- Solution iii

Solution

Multiply the first-order equation by

Ψ_{i}^{(0) *}

and integrate. The first term vanishes because Hermitian conjugation moves the unperturbed Hamiltonian onto the known bra:

⟨ Ψ_{i}^{(0)} ∣ (\hat{H}^{(0)} - E_{i}^{(0)}) ∣ Ψ_{i}^{(1)} ⟩ = ⟨ \hat{H}^{(0)} Ψ_{i}^{(0)} ∣ Ψ_{i}^{(1)} ⟩ - E_{i}^{(0)} ⟨ Ψ_{i}^{(0)} ∣ Ψ_{i}^{(1)} ⟩ = 0.

Here

E_{i}^{(0)}

is real. The remaining equation is

⟨ Ψ_{i}^{(0)} ∣ \hat{H}^{(1)} ∣ Ψ_{i}^{(0)} ⟩ - E_{i}^{(1)} ⟨ Ψ_{i}^{(0)} ∣ Ψ_{i}^{(0)} ⟩ = 0.

Using wavefunction normalization therefore yields

E_{i}^{(1)} = ⟨ Ψ_{i}^{(0)} ∣ \hat{H}^{(1)} ∣ Ψ_{i}^{(0)} ⟩ = \int Ψ_{i}^{(0) *} \hat{H}^{(1)} Ψ_{i}^{(0)} d τ .

Thus the first-order energy coefficient is the perturbation's expectation value in the unperturbed quantum state.

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