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Addition of spin angular momenta

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Addition of spin angular momenta

Consider the addition of two spin -1/2 angular momenta, S(1) and S(2)

1. How many states are there in the product basis?
2. If J = S(1) + S(2), what are the possible eigenvalues of the dot product of J and J?
3. By using the recursive algorithm construct all the Clebsch-Gordan coefficients for this problem.

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Solution Summary

This solution provides step by step calculations for states, eigenvalues, and Clebsch-Gordan coefficients.

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Total spin state of two particles with spin 1 and spin 1/2

A. Consider a system of 2 particles: particle 1 has spin 1, and particle 2 has spin 1/2. Let S be the total angular momentum operator of the two particles, where the eigenvalues of S^2 and Sz are ħ^2s(s+1) and ħms, respectively. The particles are in the state s= 3/2 and ms = 1/2.

Calculate the wave function |s = 3/2 ms = 1/2> as a linear combination of the wave functions |m1s m2s>, where m1s is the z component of the spin of particle 1, and m2s is the z component of the spin of particle 2.

b. Find the probabilities that the z component of the spin of particle 1 is
i) m1s = +1
ii) m1s = 0
iii) m1s = -1

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