A spin32 particle and a spin1 particle are confined in a box

A spin-3/2 particle and a spin-1 particle are confined in a box in a state of total spin 3/2 and total spin z-component h/2. If one measures the z-component of the spin-3/2 particle, what are the possible values and with what probability is each obtained?

Solution

Here, s1 = 3/2, s2 = 1, s = 5/2, 3/2, 1/2.

If s = 3/2, sz = 3/2, 1/2, -1/2, -3/2

To get there probabilities one has to write the state using CG coefficients in terms of the uncoupled states.

Total state with s = 3/2 is then (notation used for coupled state is ||s sz>>

|| s sz>> = a ||3/2 3/2>> + b||3/2 1/2>> + c||3/2 -1/2>> + d||3/2 -3/2>>

Each of these couped states can be written in terms of uncopled states using CG coeffcients.

THen to find the probability find <<3/2 3/2||s sz>> = a and so on.