Peculiar feature of a commutator, can anyone explain?

In summary, the conversation discusses the issue of Hermiticity of L_3 on a space of functions with finite norm on a finite interval. D. Judge has pointed out that L_3 is not Hermitian and there is a flaw in the argument. W. Louisell has suggested that using \sin(\phi) and \cos(\phi) may be more consistent than using \phi. The speaker also mentions the importance of being careful about conditional convergence when taking traces of products of infinite dimensional matrices and the need to express \phi in the l,m basis.
  • #1
haitao23
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  • #2
You've happened upon a very tricky issue indeed! I've got a couple of papers to point you to.

It has been pointed out by D. Judge1 that [itex]L_3[/itex] is not Hermetian on a space of functions that have finite norm on a finite interval. Since [itex]\phi\in(0,2\pi )[/itex], that is exactly what we are dealing with here. Since you tacitly used the Hermiticity of [itex]L_3[/itex] to evaluate those matrix elements, there is a flaw in your argument.

I didn't think of this myself, it is mentioned as a footnote in Problem 9.15 in Liboff's Introductory Quantum Mechanics. In that problem he brings up another inconsistency issue with this commutator (maybe I'll post it if you're sufficiently interested and and I'm feeling sufficiently ambitious). He mentions that W. Louisell1 has pointed out that "more consistent angle variables are [itex]\sin(\phi)[/itex] and [itex]\cos(\phi)[/itex]."

1D. Judge, Nuovo Cimento 31, 332 (1964)
2W. Louisell, Phys. Lett. 7, 60 (1993)

Hope that helps,

Tom
 
  • #3
You are taking traces of products of infinite dimensional matrices. When you do this, you must be careful about conditional convergence. If you are not using the coordinate represntation, then what do you mean by [itex]\phi[/itex]? You must first express [itex]\phi[/itex] in the [itex]l,m[/itex] basis, and then you will see that your "coordinate independent" conclusion is not so obvious.
 

Related to Peculiar feature of a commutator, can anyone explain?

1. What is a commutator and what is its purpose?

A commutator is a mechanical component found in certain types of electric motors. Its purpose is to reverse the direction of current flow in the electromagnets, which in turn causes the motor to rotate in a specific direction.

2. How does a commutator differ from a slip ring?

A commutator and slip ring both serve the same purpose of reversing current flow, but they differ in design. A commutator consists of copper segments fixed to the rotor of a motor, while a slip ring is a continuous ring with a metal brush making contact with it.

3. What is the role of brushes in a commutator?

Brushes are small carbon pieces that make contact with the commutator segments. They allow the flow of current from the external power source to the commutator, which then transfers it to the electromagnets.

4. Why is it important to have a well-functioning commutator in an electric motor?

A well-functioning commutator ensures that the motor runs smoothly and efficiently. If the commutator is damaged or malfunctioning, it can cause the motor to overheat, wear out faster, and even stop working altogether.

5. What are some common issues that can occur with a commutator?

Some common issues with a commutator include wear and tear of the segments, buildup of debris or dirt, and faulty brushes. These can lead to poor motor performance, overheating, and potential motor failure.

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