 Solution

ID: 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/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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