I was reading up on the density of states and i hit on a snag...

I dint understand how they take the "volume" of the density of states...i mean density of states corresponds to the density of energy level rite...how can u consider the "volume" of energy level??

Moreover....can you also give me the qualitative reason for the density of states function??

Its exactly what it says. Given a specific energy level E we can find the density of the states at that level per unit volume per unit energy(we have to normalize so its more useful).

Now any time your talking about density you must count something and then divide by some related factor that in some sense normalizes it. Mass density is the the mass(which is effectively the "count" of the number of atoms) divided by the volume that contains that mass. Ofcourse this is a little easier to do as we don't have to know the mass of the atoms nor how many because we have newtons laws to help us out. (we just weigh the thing and measure its volume through whatever means we can).

So in this case we have to somehow count the states of the electrons(or whatever were describing). Think of a material that has its electrons all in different states. We sorta want to have a means of knowing the density of those states in the material. Basically a proportion of the states that contribute to the total energy in the volume.

That is, we need to find all the states that contribute to energy E and then divide by the volume we looked at and divide by the total energy. Think of an atom with its electrons in different states. These states correspond to definite energy levels but what proportions correspond to what energy levels?

For a free electron we have E = (hbar*k/2/m)^2. This gives us the energy in terms of phase space.

The key here is that phase space is a space of states. Each unit in the phase space corresponds to a specific state. That means now its really easy to add up the states because we just have to integerate over a volume in it. But what volume? Well we know that its spheres because constant energy levels result in sphere in k space. Remember, if you solve shrodingers wave equation you get that psi_k is dependent on k. Think of k as an independent variable you then just have to add up the right number of psi_k's that depend on E because thats what were after. (states in terms of energy)

So take our free particle,

E = (hbar*k)^2/2/m

and then integrate over a sphere with radius k = sqrt(2mE/hbar^2). Remember, essentially we want to count all the states within our constant energy surface. In k space we just have to integrate. Since we are dealing with spheres in k space and volume tells us the number of k spaces(if say, we were dealing with a 1x1x1 = 1 cube then we would have exactly one state, if a 2x3x1 cube we would have 6 states). Therefor the number of states that give us an energy less or equal to E is

2*4/3*pi* sqrt(2mE/hbar^2)^3

which is simply the volume of a sphere in k space with radius sqrt(2mE/hbar^2). The extra factor of 2 comes in because each state is actually 2 states(spin up and spin down for electron).

So now we are counting the number of states that have at most energy E. But here we assumed that the unit length was a state when in fact its L/2/pi. (I didn't scale my axis because I forgot). This means that my volume is off by a factor of (L/2/pi)^3. (instead of counting unit cubes I have to count different sized cubes and that means I just have to multiply by the scaling factor).

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