The quantum variational principle says that every admissible nonzero trial wavefunction obeys

E = \frac{⟨ Ψ ∣ H ^ ∣ Ψ ⟩}{⟨ Ψ ∣ Ψ ⟩} \geq E_{0},

where

E_{0}

is the ground-state energy. Thus the Rayleigh quotient is an upper bound. Equality holds when the trial lies entirely in the ground-state subspace.

ii

Solution

Ψ = \sum_{n} c_{n} ϕ_{n}

, with

\hat{H} ϕ_{n} = E_{n} ϕ_{n}

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

Every summand in the numerator is nonnegative because

E_{n} \geq E_{0}

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