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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-2/31a/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 31A / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

For spin

1/2

, $S = ℏ σ /2$ . The Pauli matrices satisfy

σ_{i} σ_{j} = δ_{ij} I + i ϵ_{ijk} σ_{k}

. The antisymmetric term vanishes on contraction with

n_{i} n_{j}

, and

∣ n ∣ = 1

, so

(n \cdot σ)^{2} = I

. Separating the even and odd terms of the exponential series gives

e^{- i θ n \cdot S /ℏ} = I cos (θ /2) - i (n \cdot σ) sin (θ /2) .

For the single-particle Hamiltonian, unitary time evolution consequently acts as

∣ ↑ ⟩ \mapsto cos (α t /2) ∣ ↑ ⟩ - i sin (α t /2) ∣ ↓ ⟩, ∣ ↓ ⟩ \mapsto cos (α t /2) ∣ ↓ ⟩ - i sin (α t /2) ∣ ↑ ⟩ .

For two spins the joint eigenstates of total angular momentum are the spin-one-half triplet state

∣1, 1 ⟩ = ∣ ↑↑ ⟩, ∣1, 0 ⟩ = \frac{∣ ↑↓ ⟩ + ∣ ↓↑ ⟩}{2}, ∣1, - 1 ⟩ = ∣ ↓↓ ⟩,

with

J^{2} = 2 ℏ^{2}

and

J_{3} = m ℏ

, and the spin-one-half singlet state

∣0, 0 ⟩ = \frac{∣ ↑↓ ⟩ - ∣ ↓↑ ⟩}{2}, J^{2} = J_{3} = 0.

These follow by adding the two spin-

1/2

angular momenta, or by direct action of the Pauli matrices.

The two terms of the specified Hamiltonian commute, so its evolution factorizes into rotations with opposite angles on the two tensor factors. The unique initial

J_{3} = ℏ

state is

∣ ↑↑ ⟩

. With

c = cos (λ t /2)

,

s = sin (λ t /2)

, its evolved state is

c^{2} ∣ ↑↑ ⟩ + s^{2} ∣ ↓↓ ⟩ + i cs (∣ ↑↓ ⟩ - ∣ ↓↑ ⟩) .

The singlet amplitude is

i 2 cs

, giving the requested probability

P (J^{2} = 0) = 2 c^{2} s^{2} = \frac{1}{2} sin^{2} (λ t) .

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