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El Hombre Invisible
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Hi there. Long time no see. I hope you're all well.
An infinite 1D system has electron plane waves occupying states 0 <= E <= E_F. At time t=0, a potential step is introduced such that V=0 for x<0 and V=V' for x>0. What is the electron density when the system reaches equilibrium again?
The initial (unperturbed) electron density, in atomic units, is [tex]n(x) = \int_{0}^{k_{F}} \frac{dk}{\pi}[/tex] where [tex]k_{F} = \sqrt{2E_{F}}[/tex]
Well, when the pertubation is switched on the wavenumbers for x<0 are unchanged while those for x>0 are given by [tex]k = \sqrt{2(E - V'}[/tex]. The initial occupancy for x>0 is [tex]V' < E < E_{F}+V'[/tex]. When in equilibrium, the left and right sides must be energetically equal. Since the initial energy difference is V', and the system is symmetric about x=0, I'm figuring that the final occupancies will be:
[tex]0 < E < E_{F} + \frac{V'}{2}[/tex] for x < 0
[tex]V' < E < E_{F} + \frac{V'}{2}[/tex] for x > 0
in atomic units. The equation for the ground state depends on [tex]\sqrt{V'}[/tex], but looking at a graph the difference between n(x) on the left and right sides is just V'. So clearly I'm using the wrong equation. Anyone know the right one?
Homework Statement
An infinite 1D system has electron plane waves occupying states 0 <= E <= E_F. At time t=0, a potential step is introduced such that V=0 for x<0 and V=V' for x>0. What is the electron density when the system reaches equilibrium again?
Homework Equations
The initial (unperturbed) electron density, in atomic units, is [tex]n(x) = \int_{0}^{k_{F}} \frac{dk}{\pi}[/tex] where [tex]k_{F} = \sqrt{2E_{F}}[/tex]
The Attempt at a Solution
Well, when the pertubation is switched on the wavenumbers for x<0 are unchanged while those for x>0 are given by [tex]k = \sqrt{2(E - V'}[/tex]. The initial occupancy for x>0 is [tex]V' < E < E_{F}+V'[/tex]. When in equilibrium, the left and right sides must be energetically equal. Since the initial energy difference is V', and the system is symmetric about x=0, I'm figuring that the final occupancies will be:
[tex]0 < E < E_{F} + \frac{V'}{2}[/tex] for x < 0
[tex]V' < E < E_{F} + \frac{V'}{2}[/tex] for x > 0
in atomic units. The equation for the ground state depends on [tex]\sqrt{V'}[/tex], but looking at a graph the difference between n(x) on the left and right sides is just V'. So clearly I'm using the wrong equation. Anyone know the right one?