Charge density of an infinite 1D system

[El Hombre Invisible]

Hi there. Long time no see. I hope you're all well.

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 $$n(x) = \int_{0}^{k_{F}} \frac{dk}{\pi}$$ where $$k_{F} = \sqrt{2E_{F}}$$

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 $$k = \sqrt{2(E - V'}$$. The initial occupancy for x>0 is $$V' < E < E_{F}+V'$$. 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:

$$0 < E < E_{F} + \frac{V'}{2}$$ for x < 0
$$V' < E < E_{F} + \frac{V'}{2}$$ for x > 0

in atomic units. The equation for the ground state depends on $$\sqrt{V'}$$, 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?

 

[El Hombre Invisible]

Is there something wrong with my wording here? Please tell me if there is and I will amend the question. I could do with sussing this in the next week. Cheers... EHI