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 Incoming links: Trial wavefunction

Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 1 / A2 / a / Solution 2026-10-20

Let

{ϕ_{j}}

be a complete orthonormal eigenbasis of the Hamiltonian, with

H ϕ_{j} = E_{j} ϕ_{j}

and

E_{j} \geq E_{0}

. Expand a nonzero admissible trial wavefunction as

Ψ = \sum_{j} c_{j} ϕ_{j}

. Then

⟨ Ψ ∣ Ψ ⟩ = \sum_{j} ∣ c_{j} ∣^{2}, ⟨ Ψ ∣ H ∣ Ψ ⟩ = \sum_{j} ∣ c_{j} ∣^{2} E_{j} .

Consequently

E - E_{0} = \frac{\sum _{j} ∣ c _{j} ∣ ^{2} ( E _{j} - E _{0} )}{\sum _{j} ∣ c _{j} ∣ ^{2}} \geq 0, E \geq E_{0} .

Every numerator term is nonnegative and the denominator is positive. Equality requires the trial to lie wholly in the ground-state subspace; it need not be an individually chosen eigenvector when the ground energy is degenerate. This is the quantum variational principle, and wavefunction normalization before calculation is optional because the quotient removes the overall scale.

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