Question

One can show that the raising and lowering operators for angular momentum, \(J_\pm = J_x \pm iJ_y\), commute with J², and that, if j, m are the eigenvalues of J, J_z, then

for appropriately chosen phase conventions of the state vectors. Use these properties to express those states |j, m) for which m = l -1/2 in terms of the states |I, m_l; s, m) with s = 1/2.

Answer 1

According to the theory of angular momentum coupling, the possible values of the total angular momentum are j = l + 1/2 and j = l - 1/2 for s = 1/2.

(a) For

(b) For j = l - 1/2, let

Similarly, Eq. (2) can be written as

a² + b² = 1.

The last two equations show that a, b are both real and can be taken as