Three Spins

Consider three particles of spin 1/2 which have no motion. The raising (s⁺ = s_x + is_y) and lowering (s^- = s_x - is_y) operators of the individual spins have the property

(i)

(ii)

where the arrows indicate the spin orientation with regard to the z-direction.

a) Write explicit wave functions for the four J = 3/2 states: (M = 3/2, 1/2, -1/2, -3/2).

b) Using the definition that construct the 4 × 4 matrices which represent the J⁺ and J^- operators.

c) Construct the 4 × 4 matrices which represent J_x and J_y.

d) Construct from J_x, J_y, J_z the value of the matrix J²_.

SOLUTION

a) We use the notation that the state with three spins up is |↑₁↑₂↑₃⟩. This is the state with M = 3/2. We operate on this with the lowering operator J^-, which shows that the states |3/2, M⟩ with lower values of M are

(1)

(2)

(3)

(4)

b) From the definition of J⁺ we deduce that

(5)

(6)

(7)

(8)

The matrix J^- is the Hermitian conjugate of J⁺:

(9)

c) Because J_x = (J⁺ + J^-)/2 and J_y = -i (J⁺ - J^-)/2  we can construct

(10)

(11)

(12)

d) To find the matrix J² = J²_x + J²_y + J²_z, we square each of the three matrices and add them. This gives J² = 15Ĩ/4, where Ĩ is the 4 × 4 unit matrix. This is what one expects, since the eigenvalue of J² is J(J + 1), which is 15/4 when J = 3/2.