Matrix represntation of angular momentum operator (QM)

joker_900

1. Homework Statement
The matrix R(q) for rotating an ordinary vector by q around the z-axis is given by@

cosq -sinq 0
sinq cosq 0
0 0 1

From R calculate the matrix J(z).

2. Homework Equations
-

3. The Attempt at a Solution

All I know is that U(q) = exp[-iJ(z)q] is the unitary operator which rotates a system, and I believe that

R(q) x = U(dagger)(q) x U(q)

Where x is the position vector.

I have no idea where to go from here, other than expanding U with a taylor series but this didn't seem to go anywhere.

Thanks

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AEM

Gold Member
You will probably find page 315 and 316 of Shankar's Principles of Quantum Mechanic helpful.

joker_900

You will probably find page 315 and 316 of Shankar's Principles of Quantum Mechanic helpful.
Thanks, but I still don't see how to do the question at all. Are the pages right - do they change with editions?

AEM

Gold Member
You will probably find page 315 and 316 of Shankar's Principles of Quantum Mechanic helpful.
My edition of Shankar is from 1981. By chance I was reviewing this topic the day before your original post. I'll spend some time thinking about it and see if I can help you out.

nrqed

Homework Helper
Gold Member
1. Homework Statement
The matrix R(q) for rotating an ordinary vector by q around the z-axis is given by@

cosq -sinq 0
sinq cosq 0
0 0 1

From R calculate the matrix J(z).

2. Homework Equations
-

3. The Attempt at a Solution

All I know is that U(q) = exp[-iJ(z)q] is the unitary operator which rotates a system, and I believe that

R(q) x = U(dagger)(q) x U(q)

Where x is the position vector.

I have no idea where to go from here, other than expanding U with a taylor series but this didn't seem to go anywhere.

Thanks
I think that the correct relation is simply R(q) = exp[-i Jz q].

Just work to first order in q. Then $e^{-iJ_z q} \approx 1 - i J_z q$ (where 1 here is the unit matrix) . Replace cos(q) by 1 and sin(q) by q in the matrix R(q) and you will get the expression for Jz easily.

AEM

Gold Member
I think that the correct relation is simply R(q) = exp[-i Jz q].

Just work to first order in q. Then $e^{-iJ_z q} \approx 1 - i J_z q$ (where 1 here is the unit matrix) . Replace cos(q) by 1 and sin(q) by q in the matrix R(q) and you will get the expression for Jz easily.
I just logged on to make similar comments. Take the angle q to be very small, or infinitesimal, leads to the replacements given above.

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