Edge States in Integer Quantum Hall Effect (IQHE)

[Mr_Allod]

Homework Statement

Consider a free particle in a magnetic field bounded by a strong confining potential. Find:
a. The landau energy levels
b. The maximal and minimal values of $k$ assuming only the $n = 0$ landau level is occupied
c. The states with these $k$-values are known as edge states. Find the positions in the y-direction of these states.
b. The velocity of electrons corresponding to these states.

Relevant Equations

Hamiltonian: $H = \frac {(-i\hbar\nabla -e \vec A)}{2m} + \frac {m\omega_0^2y^2}{2}$
Landau Guage: $\vec A = (-yB,0,0)$

Hello there, I am having trouble understanding what parts b-d of the question are asking. By solving the Schrodinger equation I got the following for the Landau Level energies:

$$E_{n,k} = \hbar \omega_H(n+\frac 12)+\frac {\hbar^2k^2}{2m}\frac{\omega^2}{\omega_H^2}$$

Where $\omega_H = \sqrt{\omega^2 + \omega_c^2}$ and $\omega_c$ is the cyclotron frequency $\omega_c = \frac {eB}{m}$. At $n = 0$ this simplifies to:

$$E_{0,k} = \frac {\hbar \omega_H}{2} + \frac {\hbar^2k^2}{2m}\frac{\omega^2}{\omega_H^2}$$

Now at this point I'm not sure what exactly the question is asking. On a hunch I tried to find the roots of the expression:
$$E_{0,k}-\frac {\hbar \omega_H}{2} + \frac {\hbar^2k^2}{2m}\frac{\omega^2}{\omega_H^2}=0$$

This yielded:
$$k = i\frac {\hbar}{\sqrt{m\hbar\omega_H-2mE}}$$

Which does not seem very useful. After this point I really don't understand what the question is asking. Assuming I could find the $k$-values how would I translate these into positions in the y-direction? Or even velocities? I am thoroughly confused by this and would appreciate any help thank you!

 

[mark726]

Hello there,

Thank you for reaching out and explaining your confusion with parts b-d of the question. I can understand how the problem may seem unclear, so let me try to break it down for you.

Part b is asking you to find the possible values of k for the ground state (n = 0) energy level. In other words, you need to find the values of k that satisfy the equation you derived for E0,k. These values of k represent the allowed momentum states for an electron in the ground state.

Part c is asking you to find the corresponding positions in the y-direction for these momentum states. To do this, you can use the relationship between momentum and position in quantum mechanics, which is given by p = ħk. So, for each value of k that you found in part b, you can calculate the corresponding position in the y-direction.

Finally, part d is asking you to find the velocities for these momentum states. Again, you can use the relationship between momentum and velocity in quantum mechanics, which is given by v = p/m. So, for each value of k, you can calculate the corresponding velocity in the y-direction.

I hope this helps clarify the question for you. Please let me know if you have any further questions or if you need any more assistance. Good luck with your calculations!