Algebra of Angular Momentum

Given the commutator algebra

a) Show that J² = J²₁ + J²₂ + J²₃ commutes with J₃.

b) Derive the spectrum of {J², J₃} from the commutation relations.

SOLUTION

a)

b) Since J² and J₃ commute, we will try to find eigenstates with eigenvalues of J² and J₃ denoted by |jm⟩ where j, m are real numbers:

Since J²₁ + J²₂ + J²₃ ≥ J²₃ we know that λ ≥ m². Anticipating the result, let λ ≡ j(j + 1). Form the raising and lowering operators J₊ and J_-:

Find the commutators

(1)

From part (a) we know that [J_±, J²] = 0. We now ask what is the eigenvalue of J² for the states J_± |jm⟩:

(2)

So, these states have the same eigenvalue of J². Now, examine the eigenvalue of J₃ for these states:

(3)

In (3) we see that J_± has the effect of raising or lowering the m-value of the states |jm⟩ so that

where C^m_j± are the corresponding coefficients. As determined above, we know that j(j + 1) ≥ m², so J_± cannot be applied indefinitely to the state |jm⟩; i.e.,  there must be an m = m_max, m = m_min such that

(4)

(5)

Expand J_-J₊ and apply J_- to (4):

Either the state |jm_max⟩ is zero or j(j + 1) – m²_max – m_nax = 0. So

Similarly,

For j ≥ 0, and since m_max ≥ m_min, the only solution is

We knew that j was real, but now we have m_max – m_min = 2j = integer, so