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haitao23
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xepma said:What you proved is that the expectation value of the commutator with respect to some spherical symmetric eigenstate of L_z is zero. But this is always the case, namely, suppose if:
[itex] [A,B] = C[/itex]
For some hermitian operators A, B and a third operator C. Now take an eigenstate of the operator B, let's name it [itex]|b\rangle[/itex], such that [itex]B|b\rangle = b|b\rangle[/itex]. Then:
[itex]\langle b|[A,B]|b\rangle = \langle b|AB|b\rangle - \langle b|BA|b\rangle
=b\langle b|A|b\rangle - b\langle b|A|b\rangle = 0 [/itex]
Which is precisely what you showed, and I haven't even defined the operators yet. The fact is that we cannot conclude that the commutator is zero, because we have only considered a very particular set of expecation values. For instance, consider the following (b and b' correspond to different states):
[itex]\langle b'|[A,B]|b\rangle = b'\langle b'|A|b\rangle - b\langle b|A|b\rangle [/itex]
In general, this will not be identical to zero.
haitao23 said:How is the wavefunction in spherical coordinate defined? Is it defined with phi belonging to [0,2pi) or is it defined by imposing the condition f(phi+2pi)=f(phi)? And please give me a book (where it is to referred to also... I did not find a definition even in the dictionary like Cohen-Tannoudji book)
What I am wondering is that if the wavefunction is indeed defined with phi belonging to [0,2pi) (which in my humble opinion sounds far more natural to associate each point in space with a single point in the function) then the periodity problem of phi operator really dosen't matter as we would never get out of the 2pi domain...
haitao23 said:Hey weejee! You are really astute! I think You pinpointed the problem that has been bothering me for a month!
haitao23 said:How is the wavefunction in spherical coordinate defined? Is it defined with phi belonging to [0,2pi) or is it defined by imposing the condition f(phi+2pi)=f(phi)?
jensa said:The base manifold (space on which function is defined) is a circle, i.e. the points phi and phi+2pi are the same points, so it does not really make sense to say that the wave-function is defined outside [0,2pi). The topological constraint imposes the periodicity of the wave-functions, and any operator on this (reduced)Hilbert space must transform a periodic function to another periodic function. This is not satisfied by phi as pointed out by weejee, so it must be made periodic.
haitao23 said:So do u mean by what u said in red that the wavefunction is NOT defined outside [0,2pi)? Or from what u said in blue u seems to be defining the wavefunction outside [0,2pi) by imposing periodicity?
So after all is the wavefunction defined outside [0,2pi) ?
A commutator is a mechanical component found in electric motors and generators that helps to switch the direction of current flow within the device. This allows for the continuous rotation of the motor or generator.
A commutator consists of a set of copper segments attached to an axle. As the axle rotates, the segments come into contact with stationary carbon brushes, completing the circuit and allowing current to flow. The segments then rotate out of contact, switching the direction of the current flow.
The commutator is peculiar because it allows for the continuous rotation of the motor or generator, despite the fact that the current flow is being switched back and forth. Without the commutator, the device would only be able to rotate in one direction.
Commutators can wear down over time due to the friction between the segments and the brushes, leading to reduced efficiency and potential electrical arcing. It is important to regularly clean and maintain the commutator to prevent these issues.
Yes, there are electronic alternatives such as inverters and rectifiers that can also switch the direction of current flow in electric motors and generators. However, commutators are still widely used in many applications due to their simplicity and reliability.