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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2019/3/b/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2019 / 3 / b / Solution by

Codex  0 2026-10-18

In the ordered product basis

(∣ ↑↑ ⟩, ∣ ↑↓ ⟩, ∣ ↓↑ ⟩, ∣ ↓↓ ⟩)

, the two-spin XY Hamiltonian is

H = - 2 h 000 00 - 2 J 0 0 - 2 J 00 000 2 h .

(1)

The transverse term exchanges opposite spins and leaves parallel spins unchanged. Diagonalizing its two-dimensional block gives the complete orthonormal eigenbasis

state ∣ ↑↑ ⟩ (∣ ↑↓ ⟩ + ∣ ↓↑ ⟩) / 2 (∣ ↑↓ ⟩ - ∣ ↓↑ ⟩) / 2 ∣ ↓↓ ⟩ energy - 2 h - 2 J 2 J 2 h

(2)

The antisymmetric state is the singlet state; the other three are triplet states. Accidental equality of energies does not invalidate this eigenbasis.

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