Density of States -- alternative derivation

[Alex Cros]

I am trying to understand the derivation for the DOS, I get stuck when they introduce k-space. Why is it necessary to introduce k-space? Why is the DOS related to k-space? Perhaps if someone could come up to a slightly different derivation (any dimensions will do) that would help.
My doubt ELI5: If we have a region full of potatos and we want to find the density of potatos we count the potatos and we divide by the area/volume, we don't need to go into potato-space.

 

[DrClaude]

I am trying to understand the derivation for the DOS, I get stuck when they introduce k-space. Why is it necessary to introduce k-space? Why is the DOS related to k-space? Perhaps if someone could come up to a slightly different derivation (any dimensions will do) that would help.
My doubt ELI5: If we have a region full of potatos and we want to find the density of potatos we count the potatos and we divide by the area/volume, we don't need to go into potato-space.

The problem with that last approach is that this would give you the total density of potatoes, not the density of potatoes as a function of the distance from the middle of the field

The reason for going to k-space, part from the fact that k-space is very useful for solid state physics, is that it is easier to find the density of states in a space where the states are uniformly distributed. For particle-in-a-box states, this is the case for k-space. It would also be the case in n-space (with n the quantum number), and some authors use that instead. It would not be true for energy space, so this is why one finds the density of states first in k-space, and then converts it to energy.

If you want to find the density of states for the harmonic oscillator, then you can directly start in energy-space, since the states are uniformly distributed there (since $E \propto n$).

 

[Alex Cros]

The problem with that last approach is that this would give you the total density of potatoes, not the density of potatoes as a function of the distance from the middle of the field

The reason for going to k-space, part from the fact that k-space is very useful for solid state physics, is that it is easier to find the density of states in a space where the states are uniformly distributed. For particle-in-a-box states, this is the case for k-space. It would also be the case in n-space (with n the quantum number), and some authors use that instead. It would not be true for energy space, so this is why one finds the density of states first in k-space, and then converts it to energy.

If you want to find the density of states for the harmonic oscillator, then you can directly start in energy-space, since the states are uniformly distributed there (since $E \propto n$).

Thank you so much! Could you explain how this is done in n-space, (I can't really picture k-space) or perhaps refer to a book that does that, I am struggling to find any alternative derivation!

Thanks in advance! :)

 

[DrClaude]

Thank you so much! Could you explain how this is done in n-space, (I can't really picture k-space) or perhaps refer to a book that does that, I am struggling to find any alternative derivation!

There is not much difference between doing it in k-space and n-space.

You can find it in the R. Baierlein, Thermal Physics (CUP).

 

[ZapperZ]

Thank you so much! Could you explain how this is done in n-space, (I can't really picture k-space) or perhaps refer to a book that does that, I am struggling to find any alternative derivation!

Thanks in advance! :)

I can use the question you posed in the first post and ask you the same thing: Why would you want to do this in real space?

For solid state/condensed matter physicists, k-space is more useful for many reasons. The diffraction pattern that we get from measurements directly map the k-space of the crystal structure. But more important than that, the band structure is a set of dispersion curves in E vs. k! This band structure practically determines a significant portion of the property of the solid. And BTW, the sum of the band structure over all k-space gives the DOS at a particular energy.

We do a lot of things in k-space because it is USEFUL, more useful than in real-space.

Zz.

 

[Fred Wright]

"If we have a region full of potatos and we want to find the density of potatos we count the potatos and we divide by the area/volume, we don't need to go into potato-space."

Many years ago when I was studying the Dirac eqn. my beloved teacher, Professor Oreste Piccioni, handed back an exam paper and said to me: "Fred, the proton, she's not a little potato!"

Peace.
Fred