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Original question:

A system contains two particles: the spin of particle 1 is 3/2 and the spin of particle 2 is 1. Motion of particles can be ignored.

Part A asked to find the total spin state |5/2 3/2> using Clebsch-Gordon coefficients.

I did so, and came up with,

[tex] \mid 5/2\ \ \ 3/2\ \rangle = \sqrt{2/5} \mid 3/2\ \ \ 3/2\ \rangle\mid 1\ \ \ 0\ \rangle + \sqrt{3/5} \mid 3/2\ \ \ 1/2\ \rangle\mid 1\ \ \ 1\ \rangle [/tex]

Had no trouble with that.

Part D asks to apply the operator:

[tex]S_- = S_-^{(1)} + S_-^{(2)}[/tex],

on the composite state constructed in part A.

Now I have,

[tex] S_-\mid s\ \ m \rangle = \sqrt{s(s+1) - m(m-1)}\ \hbar\mid s\ \ m-1 \rangle [/tex]

and

[tex]S_- = S_x - iS_y[/tex]

Kinda don't know where to go from here. I understand that since we are using minus operator, spin will be decreasing. But how do values of s and m change accordingly?

A system contains two particles: the spin of particle 1 is 3/2 and the spin of particle 2 is 1. Motion of particles can be ignored.

Part A asked to find the total spin state |5/2 3/2> using Clebsch-Gordon coefficients.

I did so, and came up with,

[tex] \mid 5/2\ \ \ 3/2\ \rangle = \sqrt{2/5} \mid 3/2\ \ \ 3/2\ \rangle\mid 1\ \ \ 0\ \rangle + \sqrt{3/5} \mid 3/2\ \ \ 1/2\ \rangle\mid 1\ \ \ 1\ \rangle [/tex]

Had no trouble with that.

Part D asks to apply the operator:

[tex]S_- = S_-^{(1)} + S_-^{(2)}[/tex],

on the composite state constructed in part A.

Now I have,

[tex] S_-\mid s\ \ m \rangle = \sqrt{s(s+1) - m(m-1)}\ \hbar\mid s\ \ m-1 \rangle [/tex]

and

[tex]S_- = S_x - iS_y[/tex]

Kinda don't know where to go from here. I understand that since we are using minus operator, spin will be decreasing. But how do values of s and m change accordingly?

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