- #1

Mr_Allod

- 42

- 16

- 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!

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

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!