- #1

FermiFrustration

- 1

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- 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?

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?