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.

$$\mid j,m> =\mid\frac{5}{2},\frac{5}{2}>\: ,\: |j_1,m_1> \& |j_2,m_2>$$

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: $$|\frac{5}{2},\frac{5}{2}>=|\frac{3}{2},\frac{3}{2}>|1,1>$$

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

 

[Sekonda]

or

$$|\frac{5}{2},\frac{5}{2}>=|\frac{3}{2},\frac{3}{2},1,1>$$

 

[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!