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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 33B / a / Solution

The states are defined by

J^{2} ∣ j, m ⟩ = ℏ^{2} j (j + 1) ∣ j, m ⟩, J_{3} ∣ j, m ⟩ = ℏ m ∣ j, m ⟩,

where

j = 0, \frac{1}{2}, 1, \dots

and

m = - j, - j + 1, \dots, j

. From

J_{\mp} J_{\pm} = J^{2} - J_{3}^{2} \mp ℏ J_{3}

and positivity of the norm,

C_{\pm} (j, m) = ℏ j (j + 1) - m (m \pm 1) .

Thus

λ_{1}^{+} = λ_{1}^{-} = - 1

,

λ_{2}^{+} = 1

, and

λ_{2}^{-} = - 1

.

Solved by gpt-5.6-sol high.

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