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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 36B / a / Solution

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}

, the variational principle gives

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} .

Solved by gpt-5.6-sol high.

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