Question

A particle of spin 1/2 and a particle of spin 1 are in configuration with total spin equal to 3/2 and with the z-component of the total spin equal to 1/2 (state | 3/2 , 1/2 >). If you could measure the z-component of the spin of the spin-1 particle, what values would you get and what are the probabilities associated with each one of those values?

Answer 1

Spin of particle 1 is 1/2, hence it's z component will be -1/2 and 1/2.

Spin of particle 2 is 1, hence it's z component will be -1, 0 and 1

Note that the z components are a interval of 1 i.e. 1/2 - (-1/2) =1 and 0 - (- 1) =1 similarly for other values of z components.

The total spin can be S1+S2 or |S1-S2| , in this case since total spin is 1/2, it is S2+S1 = 1 + 1/2 = 3/2

St=3/2 thus z component can be -3/2, -1/2, 1/2, 3/2, however in this case it is 1/2.

If z component of particle with spin 1 is measured, probability is there of obtaining, -1, 0 and 1 as z components of spin.

Probability associated with each is 1/3 since each state is equally probable.