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Spin Angular Momentum

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8.6. a. A particle has s=3/2. The operator S++ is defined to be the square of the raising operator: S++ = (S+)^2, where S+ is the usual raising operator:
S+|sms = h (s(s+1) - ms(ms+1)|sms +1)^1/2

Derive the matrix corresponding to the operator S++.

b. What is the matrix corresponding to the adjoint operator (S++)?

See attachment for better symbol representation.

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This in-depth solution has step-by-step calculations to derive the matrix corresponding to the operator and the adjoint operator of S++.

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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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