Density of States for a 1D Metal at 0K - Fermi Level Calculation

[leopard]

Homework Statement

We study a one dimensional metal with length L at 0 K, and ignore the electron spin. Assume that the electrons do not interact with each other. The electron states are given by

$$\psi(x) = \frac{1}{\sqrt{L}}exp(ikx), \psi(x) = \psi(x + L)$$

$$\psi(x) = \psi(x + L)$$

What is the density of states at the Fermi level for this metal?

The Attempt at a Solution

According to my book, the total energy of the system is

$$E = \frac{\hbar^{2}\pi^{2}n^{2}}{2mL^{2}}$$

why is this?

It's evident that k = n*2*pi because of the boundary contidions. I don't know what to do next.

 

[turin]

According to my book, the total energy of the system is

$$E = \frac{\hbar^{2}\pi^{2}n^{2}}{2mL^{2}}$$

why is this?

That doesn't look right to me. What book is this? Isn't that the energy of a SINGLE electron in the energy mode n (not the energy of all of them together)?

However, I think this expression will still be useful to you, because, since you are ignoring spin, then a single electron fills an energy level, so it represents dE/dN, where N is the number of electrons in the system.