Consider a two particle system of which one particle has spin s1 and the other s2.
1. If one particle is taken from each of two sources characterized by the state vectors |s1,m1> and |s2,m2> respectively, what is the probability that the resultant two particle system will have total spin S?
2. If the particles are taken from unpolarized sources, what is the probability that the two particle system will have total spin S?
[tex]S^2|s_n,m_n \rangle = \hbar^2 s_n(s_n+1)|s_n,m_n \rangle[/tex]
The Attempt at a Solution
I need help getting started, and also I don't understand how polarization affects the nature of the problem in part 2. To calculate the probabilities don't I just construct a wavefunction for the particle in that spin state and square it? If so what goes in the bra?
[tex]| \langle \psi | s_1,s_2,m_1,m_2 \rangle |^2[/tex]
Edit: I think this is a good start:
[tex]\langle \hat S \rangle = \langle \psi|\hat S|\psi \rangle = \Sigma \langle \psi | \hat S |s_1, s_2,m_1,m_2\rangle \langle s_1, s_2,m_1,m_2|\psi \rangle [/tex]
[tex] = \hbar^2 \left [ s_1(s_1+1)+s_2(s_2+1) \right ] \Sigma |\langle s_1, s_2,m_1,m_2|\psi \rangle|^2[/tex]
But how do I quantify the squared wavefunction on the RHS? Should I simple solve for it and say that is the probability?