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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 34C / b / i / Solution

The product states

∣ n_{1} n_{2} ⟩

,

n_{a} \in {0, 1}

, are exact eigenvectors, with

H_{tot} ∣ n_{1} n_{2} ⟩ = (E_{1} n_{1} + E_{2} n_{2}) ∣ n_{1} n_{2} ⟩ .

Thus the energies attached to

∣00 ⟩, ∣01 ⟩, ∣10 ⟩, ∣11 ⟩

are respectively

0, E_{2}, E_{1}, E_{1} + E_{2} .

Solved by gpt-5.6-sol high.

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