# Inner product of two spin operators

Suppose that particle 1 (with spin 1) and particle 2 (with spin 1/2) interact via the Hamiltonian operator

H = lambda S1 dot S2 , where lambda is a constant.

Calculate the energy of the state |s,ms> .

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#### Solution Preview

You can solve this problem by considering the square of the total spin operator S = S1 + S2. We have:

S^2 = (S1 + S2)^2 = S1^2 + S2^2 + S1 dot S2 + S2 dot S1 = S1^2 + S2^2 + 2 S1 dot S2

In the last line we have used that S1 and S2 commute. This follows from the fact that S1 and S2 act on the spins ...

#### Solution Summary

The expert explains how to diagonalize the inner product of two spin operators

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