Solving Carbon Nanotube Model with Lz & Ly Expectation Values

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Discussion Overview

The discussion revolves around a problem involving a simple model of a carbon nanotube, specifically focusing on the expectation values of angular momentum components Lz and Ly in the context of an electron bound to the surface of a cylinder with a radially outpointing electric field. Participants are exploring the implications of symmetry in the system and how it relates to the expectation values of angular momentum.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant describes the setup of the problem, noting the symmetry of the system under rotation around the x-axis and questioning how this symmetry leads to the expectation values of Lz and Ly being zero.
  • Another participant argues that a non-zero expectation value for Lz or Ly would indicate a preferred direction, thus breaking the symmetry of the system.
  • A further clarification is made that the concern is specifically about non-zero expectation values rather than non-zero angular momentum components themselves, emphasizing the relationship between symmetry and expectation values.

Areas of Agreement / Disagreement

Participants appear to agree on the importance of symmetry in the system but are discussing the implications of this symmetry on the expectation values of angular momentum. The discussion remains unresolved regarding the specific physical argument needed to demonstrate that the expectation values are zero.

Contextual Notes

The discussion does not clarify the assumptions regarding the nature of the electric field or the specific mathematical treatment of angular momentum in this context, leaving some aspects open to interpretation.

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Im doing a problem with a simple model of a carbon nanotube. In this we have an electron which is bound to the surface of a cylinder from which there is a radially outpointing electric field.
We choose to put the x-axis along the axis of the cylinder such that the coordinate describing rotation around the cylinder is directly related to its angular momentum.
Now I am asked to argue that the expectation value of Lz and Ly are 0 with a physical argument. I am not sure how to approach this. Only thing I can see is that the system is symmetric under rotation around the x-axis, but I don't see where this leads me.
 
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A non-zero Lz or Ly would point in some specific direction, and ruin your symmetry.
 
but this isn't exactly as non-zero Lz or Ly it is a non-zero expectation value.. I don't completely understand I must admit - can you elaborate? :)
 
I meant a non-zero expectation value in my post. This would break symmetry: You would get a preferred direction (the direction of the spin expectation value) instead of your symmetric setup.
 

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