 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015/12/iii/solution

Each Pauli matrix is Hermitian and the coefficients

α_{j}

are real, so

T^{†} = T

. Taking the conjugate transpose of the absolutely convergent matrix exponential series yields

U^{†} = e^{- i T /2} .

- i T /2

and

i T /2

commute. Therefore

U^{†} U = e^{- i T /2 + i T /2} = I

, proving that $U$ is unitary.

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