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Addition of angular momenta problem

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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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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.

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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 ...

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