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Addition of angular momenta problem

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

Solution Preview

a. You can compute the total spin states |s, m_s> by considering the total spin state with the largest possible (or lowest possible) z component, i.e. m_s = s (or m_s = -s) and then repeatedly applying lowering (or raising operators):

S^{±} = S_x ± i S_y = S1_x + S2_x ± i S1_y ± i S2_y = S1^{±} + S2^{±}

If we can express the |s, m_s= s> state in terms of the spin states of the individual particles then we simply have to apply S^{-} on both sides of the equation repeatedly. We know how S^{-} acts on a total spin state and since this is also S1^{-} + S2^{-} we know what it does when applied to product states consisting of the individual spin states of the particles. In general, you have to find |s, m_s= s> by expressing it as an unknown superposition of product states and then applying the raising operator S^{+}. This must yield zero which gives you an equation for the ...

Solution Summary

We explain in detail how to construct the total spin states for the case where one particle is a spin 1/2 particle and the other is a spin 1 particle.