Density of states for a free electron

[kottur]

Homework Statement

1. Find the density of orbitals (often called 'density of states') for a free electron gas in
one dimension, in a box of length L.

2. Find the density of orbitals for a free electron gas in two dimensions, in a box with
area A. Compare with the three dimensional case, eqn. (19) on page 187.

eqn. (19): $D(\epsilon)=\frac{V}{2\pi^{2}}\left(\frac{2m}{h^{2}}\right)^{\frac{3}{2}}\epsilon^{\frac{1}{2}}$

Homework Equations

The one above for the second problem because I can't use it for a one dimensional box because it's got V (Volume in it).

The Attempt at a Solution

I'm not sure what equation I should be using, also I'm confused of Fermi and Bose. My professor told me there are two ways to find density of states, one with Fermi and one with Bose (if I understood correctly).

Any help is much appreciated! I really want to understand this! :)

 

[vela]

There is no single equation you should be using. Your text probably derived the density of states in three dimensions for you. The problem is asking you to do the analogous derivations for the one-dimensional and two-dimensional cases. The idea is to make you think through what they did and apply those same techniques so that you'll understand where the final formula came from. So the place to start is to look up the derivation in your book.

The difference between fermions and bosons are how they fill up the available states, which will in turn affect the density of states. How do fermions and bosons behave differently?

 

[kottur]

Fermions have "half" spins and bosons have "whole" spins. I think I need to use fermions because my textbook uses it to derive the formula for 3D DOS.

 

[vela]

You're right about the spins, but what does that have to do with how they fill up states?

 

[paranormal]

I would first clarify that the density of states refers to the number of states (or orbitals) per unit energy range. This is an important concept in understanding the behavior of electrons in a material.

For the first problem, we can use the one-dimensional version of eqn. (19) by substituting the volume V with the length L, since we are dealing with a one-dimensional box. This gives us:

D(\epsilon)=\frac{L}{2\pi^{2}}\left(\frac{2m}{h^{2}}\right)^{\frac{1}{2}}\epsilon^{\frac{1}{2}}

For the second problem, we can use the two-dimensional version of eqn. (19) by substituting the volume V with the area A, since we are dealing with a two-dimensional box. This gives us:

D(\epsilon)=\frac{A}{2\pi^{2}}\left(\frac{2m}{h^{2}}\right)^{\frac{3}{2}}\epsilon^{\frac{1}{2}}

We can compare this with the three-dimensional case, which is given by eqn. (19) on page 187, as stated in the problem. From these equations, we can see that the density of states decreases as the dimensionality of the box decreases. This is because in lower dimensions, there are fewer possible states for the electrons to occupy.

As for the Fermi and Bose statistics, these are different ways of describing the behavior of particles at the quantum level. Fermi particles, such as electrons, follow the Pauli exclusion principle, which states that no two particles can occupy the same state. Bose particles, on the other hand, can occupy the same state, which leads to phenomena such as superconductivity and superfluidity.

In summary, the density of states is an important concept in understanding the behavior of electrons in a material, and it depends on the dimensionality of the system. The Fermi and Bose statistics are different ways of describing the behavior of particles at the quantum level.