Solution

The coupled basis consists of the triplet

∣1, 1 ⟩ = ∣ + + ⟩, ∣1, 0 ⟩ = \frac{∣ +- ⟩ + ∣ -+ ⟩}{2}, ∣1, - 1 ⟩ = ∣ - - ⟩,

and the singlet

∣0, 0 ⟩ = \frac{∣ +- ⟩ - ∣ -+ ⟩}{2} .

These equations list all nonzero Clebsch--Gordan coefficients, each coefficient being

1

1/ 2

, or

- 1/ 2

as displayed, and

I = \sum_{m = - 1}^{1} ∣1, m ⟩ ⟨ 1, m ∣ + ∣0, 0 ⟩ ⟨ 0, 0∣.

Because each irreducible total-spin representation occurs once, an operator commuting with total angular momentum is

O = α P_{j = 1} + β P_{j = 0} .

In the product basis, the nonzero matrix elements are

⟨ + + ∣ O ∣ + + ⟩ = ⟨ - - ∣ O ∣ - - ⟩ = α,

⟨ + - ∣ O ∣ + - ⟩ = ⟨ - + ∣ O ∣ - + ⟩ = \frac{α + β}{2},

and

⟨ + - ∣ O ∣ - + ⟩ = ⟨ - + ∣ O ∣ + - ⟩ = \frac{α - β}{2} .

Solved by gpt-5.6-sol high.

 Ancestors (11)