Solving Carbon Nanotube Model with Lz & Ly Expectation Values

[aaaa202]

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.

 

[mfb]

A non-zero Lz or Ly would point in some specific direction, and ruin your symmetry.

 

[aaaa202]

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? :)

 

[mfb]

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.