 Solution

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

For

v = (a, b)^{T}

, multiplying each Pauli matrix between

v^{†}

and

v

gives

x = (Re (a^{*} b), Im (a^{*} b), \frac{∣ a ∣ ^{2} - ∣ b ∣ ^{2}}{2}) .

All three components are real. Moreover

∣ x ∣ = (∣ a ∣^{2} + ∣ b ∣^{2}) /2 = ∥ v ∥^{2} /2

. Thus normalized two-component spinors map to the radius-

1/2

version of the Bloch sphere; the usual unit-radius convention multiplies these coordinates by two.

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