Matrix of angular momentum operator

1. Nov 21, 2007

TURK

as known to all, we can find a matrix representation for every operator in quantum mechanics.

for example for total angular momentum of one particle j(square) the elements are j(j+1)(square)h(bar) δmm'

However I have stucked in two particle systems.

for example I could not find the matrix of j1+j2- (this is a product) here j1+ is the raising operator for first particle and j2- is the lowering operator for second one.
normally for one particle raising angular momentum operator gives the eigen value (squareroot)[j(j+1)-m(m+1)].
but in this case as far as i know, i have to find the matrix representration of product of this two operator. but for the below conditions I could not create a matrix.
lets say j1=2 j2=1 and the restriction is m= m1 +m2 = 2. that is m1 can take values 2,1 and coresponding m2 values are 0 and 1.

2. Nov 21, 2007

Count Iblis

You must first write down your basis states, e.g. you can take the product states

|1> = |j1=2, m1=2>|j2=1,m2=0>

|2> = |j1=2, m1=1>|j2=1,m2=1>

If we put A = J1^{+} J2^{-}, then the matrix elements are

A_{i,j} = <i|A|j>

e.g.

A_{1,2} =

<|j1=2, m1=2|<j2=1,m2=0|J1^{+} J2^{-}
|j1=2, m1=1>|j2=1,m2=1>

We have:

J1^{+} J2^{-}|j1=2, m1=1>|j2=1,m2=1> =

(J1^{+} |j1=2, m1=1>) (J2^{-}|j2=1,m2=1>) =

2sqrt(2)h-bar^2|j1=2, m1=2>|j2=1, m2=0>

And we see that A_{1,2} = 2sqrt(2)h-bar^2

3. Nov 21, 2007

TURK

is that a diagonal matrix or an off diagonal matrix.
you took the state 1 and state 2 to form the matrix of the operator A and after you applied the operators to the state of 2 you got the state of 1 then the kronecker delta gave you what? a diagonal matrix or what?

4. Nov 21, 2007

TURK

or could you just write the elemts of this 2x2 matrix.
thanks alot.

5. Nov 21, 2007

TURK

ah okey just understood
thnaks very much for your kindness.