Matrix represntation of angular momentum operator (QM)

[joker_900]

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).

Homework Equations

-

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

 

[AEM]

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]

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 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).

Homework Equations

-

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]

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.