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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-57/3/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 57 / 3 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The Pauli matrices obey

X ∣0 ⟩ = ∣1 ⟩

,

X ∣1 ⟩ = ∣0 ⟩

,

Y ∣0 ⟩ = i ∣1 ⟩

, and

Y ∣1 ⟩ = - i ∣0 ⟩

. Thus

X \otimes X

and

Y \otimes Y

contribute opposite amplitudes on

∣00 ⟩, ∣11 ⟩

and equal amplitudes on

∣01 ⟩, ∣10 ⟩

. For the XY exchange interaction

E = (X \otimes X + Y \otimes Y) /2

,

E ∣00 ⟩ = 0, E ∣01 ⟩ = ∣10 ⟩, E ∣10 ⟩ = ∣01 ⟩, E ∣11 ⟩ = 0.

Equivalently

E = ∣01 ⟩ ⟨ 10∣ + ∣10 ⟩ ⟨ 01∣

. It transfers an excitation across a link without creating or destroying excitations, so a sum of such links preserves the single-excitation subspace.

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