Momentum/angle values of eigenstates in a graphene system

In summary, to calculate the expectation value of the momentum in x- and y- directions and deduce the angle associated with each eigenstate, you will need to first calculate the expectation values of the momentum operators and then use the formula $\theta = \tan^{-1} \left(\frac{P_y}{P_x}\right)$.
  • #1
FermiDIrac19
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I am trying to differentiate between states in a quantum system of graphene with open boundaries in one direction and periodic boundaries in the other direction.
The eigenstates have been simulated with the eigenenergies, but how would I proceed in calculating the expectation value of the momentum in x- and y- direction in order to deduce an angle associated with each eigenstate?
 
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  • #2
To calculate the expectation value of the momentum in x- and y- directions, you will need to first calculate the expectation values of the momentum operators $\hat{P}_x$ and $\hat{P}_y$. These operators can be expressed as follows:$$\hat{P}_x = \frac{\hbar}{i} \left(\frac{\partial}{\partial x}\right)$$ $$\hat{P}_y = \frac{\hbar}{i} \left(\frac{\partial}{\partial y}\right)$$Once you have these operators, you can then calculate the expectation values for each eigenstate by taking the inner product of the state with the operator, i.e. $$\langle \psi | \hat{P}_x | \psi \rangle$$$$\langle \psi | \hat{P}_y | \psi \rangle$$where $\psi$ is the eigenstate and $\hat{P}_x$ and $\hat{P}_y$ are the momentum operators. Once you have the expectation values for each eigenstate, you can then calculate the angle associated with each eigenstate by using the following formula:$$\theta = \tan^{-1} \left(\frac{P_y}{P_x}\right) $$where $P_x$ and $P_y$ are the expectation values of the momentum in x- and y- directions for the given eigenstate.
 

Related to Momentum/angle values of eigenstates in a graphene system

1. What is momentum in a graphene system?

Momentum in a graphene system refers to the measure of the motion of electrons within the material. It is a vector quantity that describes both the direction and speed of the electron's movement.

2. How are momentum and angle values related in eigenstates?

In eigenstates, the momentum and angle values are related through the concept of Bloch's theorem. This theorem states that the momentum and angle values of an electron in a crystalline material, such as graphene, are related by a periodic function.

3. What is the significance of eigenstates in a graphene system?

Eigenstates in a graphene system are important because they represent the allowed energy states of the electrons within the material. These states determine the electronic and optical properties of graphene, making them crucial for understanding its behavior.

4. How do the momentum and angle values of eigenstates affect the conductivity of graphene?

The momentum and angle values of eigenstates have a direct impact on the conductivity of graphene. In fact, the conductivity of graphene is highly anisotropic, meaning it varies with direction, due to the unique momentum and angle values of its eigenstates.

5. Can the momentum and angle values of eigenstates be manipulated in a graphene system?

Yes, the momentum and angle values of eigenstates can be manipulated in a graphene system through the application of external forces, such as an electric or magnetic field. This manipulation can lead to changes in the electronic and optical properties of graphene, making it a promising material for future technological applications.

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