Calculating clebsch gordan coefficients

[indecicia]

Homework Statement

A particle of spin 1 and a particle of spin 2 are at rest in a configuration such that the total spin is three and its z component = 1. If you measured the z component of the ang mom of the spin 2 particle, what values might you get and what probabilities for each?

Homework Equations

I am trying to calculate the probabilities, not look them up in a table. I *think* the relevant equation is the recursion relation:
sqrt((j+-m)(j+-m+1))<j1j2;m1m2|j1j2;j,m+-1> = (stuff for changing m1, which i think i can disregard) + sqrt((j2-+m2+1)(j2+-m2))<j1j2;m1,m2-+1|j1j2;jm>
+- means plus sign over minus sign

The Attempt at a Solution

ok, so looking at possible cominations, m2 = 0, +1, +2
now, i think the key is plugging the right numbers into the square root, squaring it to get probability, adding up the three probabilities and normalizing them.
For measuring m2 = 0, i plugged in m2 = 1 and used the upper sign because i want m2-+1 to be equal to zero. i got sqrt(6). i think this is right, because i know the answer from looking at a table. but for m2 = 0 i plugged in 1 again but this time took the lower sign and i got sqrt(4) which is not right. so now i wonder about my whole method.

 

[indecicia]

I'm still stuck

 

[malawi_glenn]

do you have to use the recursion method? Can you use ladder operators instead?

 

[indecicia]

Thanks for you reply. I actually figured out the above one of the 2 problems last night by using ladder operators and then normalizing. Hopefully I can figure out how to use ladder operators to do the second one.

 

[malawi_glenn]

Thanks for you reply. I actually figured out the above one of the 2 problems last night by using ladder operators and then normalizing. Hopefully I can figure out how to use ladder operators to do the second one.

let J = s1 + s2 = 1+2

Then construct all J = 3 states, then you know that the state with J=2, mJ=2 is orthogonal to state J=3, mJ=2, then you have the starting point for making all J=2 states. and so on.