Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 3 / c

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- Solution c

Solution

In the defining representation,

H = (10 0 - 1), E_{+} = (0010), E_{-} = (0100) .

Multiplying the three factors in the disentangling identity and comparing with

e^{i θ σ_{1} /2}

gives

α = γ = tan \frac{θ}{2}, exp (\frac{i β H}{2}) = (cos (θ /2) 0 0 sec (θ /2)) .

This factorization is the coordinate patch where

cos (θ /2) \neq = 0

.

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