Question

Determine the number of combined states, in the form of total |s m s >, for the combination of four spin ½ particles.

Do NOT determine the coefficients in the composite states.

Hint: make two combinations of spin - ½ particles, one for spins 1 and 2, and one for spins 3 and 4, and combine those

Answer 1

Here, particle 1, 2, 3, 4 means particles A, B, C, D respectively.

So, the total number of states = 2 * (number of states for s=0) + 3 * (number of stats for s=1) + 1 * (number of states for s=2)

   = 2 * 1 + 3 * 3 + 1 * 5

   = 2 + 9 + 5

   = 16

Note: If you do not look at the total spin and look at there individual product states in the form of | s₁, m_s1, s₂, m_s2, s₃, m_s3, s₄, m_s4>    then there will be 2⁴ = 16 states. So, everything is consistent.

Do comment for further clarification, if needed.