What Are the Quantum Angular Momentum States for a Pair of Spin-1/2 Particles?

[philip041]

A pair of spin half particles in a bound state. Spin angular momentum operators S1 and S2. In l= 0, no orbital component, the operator for total angular momentum is J = S1+S2

Possible ang mom states can be denoted by |J, M> where J and M are quantum numbers for J^2 and Jz respectively. Write down the two possible values of J and corresponding eigenvalues of J^2. For each J list allowed M.

Answer

J^2 eigenvalue for a given J is J(J+1)hbar^2. The two possible values of are J=1 eigenvalue 2hbar^2 and J=0 eigenvalue 0



Sorry long question! Why is J = 1 and J=0. Couldn't it also be J=-1? Or can S1 and S2 only be 1/2 or 0?

Cheers

 

[latentcorpse]

you just said two spin half particles. therefore s1 and s2 are either +1/2 or -1/2
but we add them using the addition of angular momentum theorem

classically you just add them and the possibilities are -1,0 or 1

quantum mechanically j takes values ranging from |s1|+|s2|,...,||s1|-|s2||

so if s1,s2 are either 1/2 or -1/2 then |s_i|= 1/2
and so j ranges from 1 to 0