Solution

Contracting the result of part (c) with the Pauli matrices gives

[T^{a}, T^{b}] = \frac{1}{4} σ_{ij}^{a} σ_{k l}^{b} (δ_{jk} T_{i l} - δ_{i l} T_{kj}) = \frac{1}{4} [σ^{a}, σ^{b}]_{i l} T_{i l} = i ε_{ab c} T^{c} .

Thus the

T^{a}

su (2)

commutation relations. The second Pauli matrix gives

T^{2} = \frac{i}{2} (a_{y}^{†} a_{x} - a_{x}^{†} a_{y}),

and therefore

L = 2 T^{2} .

On the energy level

E = N + 1

, the total occupation is

N

. The

N + 1

states

∣ n_{x}, N - n_{x} ⟩

form the spin-

j = N /2

irreducible representation. Every component of its angular momentum generator has eigenvalues

m = - \frac{N}{2}, - \frac{N}{2} + 1, \dots, \frac{N}{2} .

Since

L = 2 T^{2}

, the possible orbital angular momenta are

ℓ = - N, - N + 2, \dots, N - 2, N .

There are

N + 1

distinct values, exactly matching the degeneracy found in part (a); hence each occurs once. Restoring

ℏ

multiplies every

ℓ

ℏ

Solved by gpt-5.6-sol high.

 Ancestors (11)