Combined system state as product of states

Sekonda

Hey,

I have to express the combined system state of j=5/2, m=5/2 in terms of the products of states j1,m1 and j2,m2.

[tex]\mid j,m> =\mid\frac{5}{2},\frac{5}{2}>\: ,\: |j_1,m_1> \& |j_2,m_2>[/tex]

I know that one way of achieving this is for j1=3/2 and j2=1 but I'm not sure how to express this - I think this is involving Clebsch-Gordon coefficients.

Thanks guys,
SK

DocZaius

Just from inspecting your J and M, it seems clear a (spin 3/2 with max m=3/2) and a (spin 1/2 with max m=1/2) would combine to that state. As a rule, if your M happens to be the sum of two particular spins' max m, then you have a straightforward product of the states at their max m.

Sekonda

I suppose I'm confused in how I could write that the 5/2, 5/2 state was the same as 3/2, 3/2 and a 1,1.

Would this simply be: [tex]|\frac{5}{2},\frac{5}{2}>=|\frac{3}{2},\frac{3}{2}>|1,1>[/tex]

I'm not really sure what is meant by the product of two states - what notation would be used.

Sekonda

or

[tex]|\frac{5}{2},\frac{5}{2}>=|\frac{3}{2},\frac{3}{2},1,1>[/tex]

DocZaius

I've always used the first one, myself. Not sure about the second.

Sekonda

Right cool, I thought so to but I'm just a bit confused with my notes - I have two very similar way of writing it. Cheers.

DocZaius

Also, I should add another condition to my rule above.

As a rule, if your M happens to be the sum of two particular spins' max m and J=M, then you have a straightforward product of the states at their max m. This holds true if you replace instances of "max" with min" in the previous sentence.

Sekonda

Indeed, this makes sense. Thanks for the help DocZaius!