Matrix represntation of angular momentum operator (QM)

In summary, the angular momentum operator in quantum mechanics is a vector operator that represents the rotational motion of a particle in three-dimensional space. It is represented by a matrix in quantum mechanics, which has the properties of being Hermitian, having quantized eigenvalues, and satisfying commutation relations with position and momentum operators. This matrix representation is used to calculate possible values of angular momentum, find eigenstates and eigenvalues, and determine probabilities of measurement outcomes. It is a crucial concept in understanding the behavior of particles with angular momentum and has applications in various fields of physics.
  • #1
joker_900
64
0

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
 
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  • #2
You will probably find page 315 and 316 of Shankar's Principles of Quantum Mechanic helpful.
 
  • #3
AEM said:
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?
 
  • #4
AEM said:
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.
 
  • #5
joker_900 said:

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 [itex] e^{-iJ_z q} \approx 1 - i J_z q [/itex] (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.
 
  • #6
nrqed said:
I think that the correct relation is simply R(q) = exp[-i Jz q].

Just work to first order in q. Then [itex] e^{-iJ_z q} \approx 1 - i J_z q [/itex] (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.
 

Related to Matrix represntation of angular momentum operator (QM)

1. What is the angular momentum operator in quantum mechanics?

The angular momentum operator in quantum mechanics is a mathematical representation of the angular momentum of a particle. It is a vector operator that describes the rotational motion of a particle in three-dimensional space.

2. How is angular momentum represented in matrix form in quantum mechanics?

In quantum mechanics, the angular momentum operator is represented by a matrix, which is a mathematical object that represents a linear transformation. The matrix representation of angular momentum operator is obtained by expressing the vector operator in terms of the position and momentum operators.

3. What are the properties of the matrix representation of angular momentum operator?

The matrix representation of angular momentum operator has the following properties:

  • It is Hermitian, meaning its transpose is equal to its complex conjugate.
  • Its eigenvalues are quantized, meaning they can only take on certain discrete values.
  • It satisfies the commutation relations with the position and momentum operators.

4. How is the matrix representation of angular momentum operator used in quantum mechanics?

The matrix representation of angular momentum operator is used to calculate the possible values of angular momentum for a given system. It is also used to find the corresponding eigenstates and eigenvalues, which can then be used to determine the probabilities of different outcomes of a measurement.

5. What is the significance of the matrix representation of angular momentum operator in quantum mechanics?

The matrix representation of angular momentum operator is an essential tool in quantum mechanics as it allows us to describe and understand the behavior of particles with angular momentum. It is also a fundamental concept in the study of atomic and molecular structure, as well as other areas of physics such as nuclear physics and solid-state physics.

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