Calculating Density of States in One-Dimensional Metals at 0K

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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

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

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

The Attempt at a Solution

The total energy of the system is $$E = \frac{\hbar^{2}\pi^{2}n^{2}}{2mL^{2}}$$ where n is the square of the sums of the three quantum numbers that determine each quantum state.

At a certain energy all states up to $$E_{F}(0)=E_{0}n^{2}_{F}$$ is filled.

 

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