Current through Ballistic 2DEG Channel

[FermiFrustration]

Homework Statement

See attachment!
Find current through ballistic 2DEG channel assuming a parabolic potential in the channel

Relevant Equations

Schrödinger Equation, Airy's Equation, I = N (2e^2)/h V

So I am a bit uncertain what approach is best for solving this problem and how exactly I should approach it, but my strategy right now is:

1. Solve the time-independent Schrödinger Equation with the given Hamiltonian and find energy eigenvalues of system:
-Here I struggle a bit with actually solving it; if my approach is right this should be the crux of the problem
-Since the y and x-dependent parts of the Schrödinger equation are possible to separate I think it is possible to solve this as two differential equations with only one variable like:

$( -\frac{h^2}{2m^{*}}(\frac{d^2}{dx^2} + \frac{d^2}{dy^2}) + \frac{1}{2}m^* \omega^2 y^2 - E )\Psi = 0$
$(-\frac{h^2}{2m^{*}}\frac{d^2}{dx^2} - E )\Psi = (\frac{h^2}{2m^{*}}\frac{d^2}{dy^2} - \frac{1}{2}m^* \omega^2 y^2)\Psi$
$\implies ¨(-\frac{h^2}{2m^{*}}\frac{d^2}{dx^2} - E )\Psi = G$
$\implies (\frac{h^2}{2m^{*}}\frac{d^2}{dy^2} - \frac{1}{2}m^* \omega^2 y^2)\Psi = G$

-I am however unsure how to reassemble this into a complete solution.

2. Find energy eigenstates below the fermi energy - their number should be the number of available channels through the channel

3. Plug this new obtained N into I = N (2e^2)/h V with the given voltage from source to drainIs my approach right and how should I go about solving the differential equation?

 

[Nate810]

Your approach is close, but you need to solve the two-dimensional Schrödinger equation with the given parameters, rather than separating it into two equations. The two-dimensional version of the Schrödinger equation is:$( -\frac{h^2}{2m^{*}}(\frac{d^2}{dx^2} + \frac{d^2}{dy^2}) + \frac{1}{2}m^* \omega^2 (x^2+y^2) - E )\Psi = 0$There are several methods for solving this equation, such as separation of variables, Green's function techniques, and numerical methods. Once you have solved the equation to obtain the energy eigenvalues, you can then use those values to calculate the number of available channels through the channel, which you can then plug into I = N(2e^2)/h V with the given voltage from source to drain.