 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/10/a/iii/solution

Assume a complete orthonormal eigenbasis and a self-adjoint operator bounded below, with the trial state in its operator domain or associated energy-form domain. The expansion and orthogonality give

⟨ ϕ, ϕ ⟩ = \sum_{n} ∣ a_{n} ∣^{2}, ⟨ ϕ, H ϕ ⟩ = \sum_{n} E_{n} ∣ a_{n} ∣^{2} .

Consequently

F [ϕ] = \frac{\sum _{n} E _{n} ∣ a _{n} ∣ ^{2}}{\sum _{n} ∣ a _{n} ∣ ^{2}} \geq E_{0} .

It is a weighted average of eigenvalues, all at least

E_{0}

. This is the Rayleigh-Ritz variational principle; equality holds precisely when the trial state lies in the ground-state subspace. A continuous part of the spectrum would require the corresponding spectral integral in place of part of the sum.

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