Solving Carbon Nanotube Model with Lz & Ly Expectation Values

In summary, the conversation discusses a problem involving a simple model of a carbon nanotube and an electron bound to its surface. The x-axis is chosen to be along the cylinder's axis, and the expectation values of Lz and Ly are argued to be 0 with a physical argument. It is noted that a non-zero expectation value would break the system's symmetry.
  • #1
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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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  • #2
A non-zero Lz or Ly would point in some specific direction, and ruin your symmetry.
 
  • #3
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? :)
 
  • #4
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.
 

1. What is a carbon nanotube model?

A carbon nanotube model is a theoretical representation of a carbon nanotube, which is a cylindrical structure made up of carbon atoms arranged in a hexagonal lattice. This model is used to study the physical and chemical properties of carbon nanotubes and understand their behavior.

2. What is the significance of solving carbon nanotube model with Lz & Ly expectation values?

The Lz and Ly expectation values represent the angular momentum and the linear momentum of the carbon nanotube, respectively. By solving the carbon nanotube model with these expectation values, we can obtain important information about the nanotube's properties, such as its structural stability, electronic properties, and mechanical behavior.

3. How is the Lz & Ly expectation values calculated in the carbon nanotube model?

The Lz and Ly expectation values are calculated using quantum mechanical operators, which represent the angular momentum and the linear momentum of the nanotube. These operators act on the wavefunction of the nanotube, resulting in the expectation values that describe its properties.

4. What is the importance of studying carbon nanotube models?

Studying carbon nanotube models allows us to gain a better understanding of the fundamental properties and behavior of these structures. This knowledge is crucial for developing new applications and technologies that utilize carbon nanotubes, such as in electronics, energy storage, and biomedical applications.

5. How do the Lz & Ly expectation values affect the properties of a carbon nanotube?

The Lz and Ly expectation values determine the angular and linear momentum of the carbon nanotube, which in turn influences its structural, electronic, and mechanical properties. For example, a high Lz value indicates a high degree of rotational symmetry in the nanotube, while a high Ly value suggests a high degree of linear movement.

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