Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section B / 33 / a

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Solution

For a self-adjoint Hamiltonian bounded below, every nonzero admissible trial wavefunction satisfies the quantum variational principle

E [ψ] = \frac{⟨ ψ ∣ H ^ ∣ ψ ⟩}{⟨ ψ ∣ ψ ⟩} \geq E_{0} .

Here

E_{0}

is the lowest exact energy, and the trial function must obey the physical boundary conditions and have finite relevant integrals. Equality holds for a wavefunction in the ground-state eigenspace. The quotient is unchanged by normalization or any nonzero overall scalar.

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