33B (Principles of Quantum Mechanics)

j = ∣ j_{1} - j_{2} ∣, ∣ j_{1} - j_{2} ∣ + 1, \dots, j_{1} + j_{2} .

For each such

j

, measurement along any axis has magnetic quantum number

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

b

For two equal spins, exchange acts on the coupled state

∣ j, m ⟩

(- 1)^{2 j_{1} - j}

. It is therefore

- 1

when

j = 2 j_{1} - 1

, making every state in that multiplet antisymmetric, and

+ 1

when

j = 2 j_{1} - 2

, making the next multiplet symmetric.

Parity conservation and the negative parity of

X

L

. Since the two

Y

particles are identical bosons, their spin state must then be antisymmetric, so their total spin is

S = 1

L = S = 1

Conditioning on spin projection

m_{S} = 1

leaves orbital projection

m_{L} = - 1

. The angular density is therefore proportional to

∣ Y_{1, - 1} (θ, φ) ∣^{2} \propto sin^{2} θ

. Including the solid-angle measure, the requested probability is

\frac{\int _{π /4}^{3 π /4} s i n ^{3} θ d θ}{\int _{0}^{π} s i n ^{3} θ d θ} = \frac{5 2}{8} .

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