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Combined system state as product of states

by Sekonda
Tags: combined, product, state, states
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Sekonda
#1
Dec3-12, 12:14 PM
P: 209
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
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DocZaius
#2
Dec3-12, 12:28 PM
P: 293
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
#3
Dec3-12, 12:34 PM
P: 209
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
#4
Dec3-12, 12:35 PM
P: 209
Combined system state as product of states

or

[tex]|\frac{5}{2},\frac{5}{2}>=|\frac{3}{2},\frac{3}{2},1,1>[/tex]
DocZaius
#5
Dec3-12, 01:09 PM
P: 293
I've always used the first one, myself. Not sure about the second.
Sekonda
#6
Dec3-12, 01:11 PM
P: 209
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
#7
Dec3-12, 01:15 PM
P: 293
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
#8
Dec3-12, 04:18 PM
P: 209
Indeed, this makes sense. Thanks for the help DocZaius!


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