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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2025/paper-1/a1/b/iii/solution

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Past exam of the chemistry A course of the University of Cambridge / 2025 / Paper 1 / A1 / b / iii / Solution by

Codex  0 2026-10-20

The quantum variational principle states that the energy expectation of an admissible trial wavefunction is at least the ground state energy:

\frac{⟨ f , H f ⟩}{⟨ f , f ⟩} \geq E_{0} .

(1)

For the discrete, complete orthonormal basis assumed here, expand in Hamiltonian operator eigenfunctions,

f = \sum_{n} c_{n} ϕ_{n}

, with

H ϕ_{n} = E_{n} ϕ_{n}

and

E_{n} \geq E_{0}

. Then

\frac{⟨ f , H f ⟩}{⟨ f , f ⟩} - E_{0} = \frac{\sum _{n} ∣ c _{n} ∣ ^{2} ( E _{n} - E _{0} )}{\sum _{n} ∣ c _{n} ∣ ^{2}} \geq 0.

(2)

Every summand is nonnegative. Equality holds precisely when all occupied components lie in the ground eigenspace. Thus minimizing trial energies supplies upper bounds on

E_{0}

, with exact equality if the trial family contains a ground state.

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