 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/36b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

ψ_{1}

be the normalized ground state for

H_{1} = T + V_{1} .

Using it as a trial state for

H_{2} = T + V_{2}

E_{2} \leq ⟨ ψ_{1}, H_{2} ψ_{1} ⟩ = E_{1} + ⟨ ψ_{1}, (V_{2} - V_{1}) ψ_{1} ⟩ .

The pointwise assumption

V_{2} \leq V_{1}

makes the last expectation nonpositive. Hence

E_{2} \leq E_{1} .

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