 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-1/section-a/a3/solution

S_{z}

(α, β)

, the spin one-half operator is

S_{x} = (ℏ/2) (0110)

a α + b β

s_{x}

obeys

b = (2 s_{x} /ℏ) a

and

a = (2 s_{x} /ℏ) b

. A nonzero vector requires

(2 s_{x} /ℏ)^{2} = 1

. Choosing normalized quantum phases gives

s_{x} = + \frac{ℏ}{2}, χ_{+} = \frac{α + β}{2}; s_{x} = - \frac{ℏ}{2}, χ_{-} = \frac{α - β}{2} .

These spin eigenstates are orthogonal and have unit norm because

α

and

β

do.

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