- #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!