The density of states independent of Boundary Conditions

[dRic2]

TL;DR Summary

Are you familiar with a more or less rigorous argument that proves the independence of the density of states for a gas of non interacting particles form its surroundings?

Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk particles. These books then proceed to show the equivalence for the specific case of periodic boundary conditions and the "particle-in-a-box"-like boundary conditions. I like this intuitive approach, but I would like to get at least a more mathematical intuition of why all of this works.

I also noticed that the famous density of states for a free particle $\frac V {(2 \pi)^3}$ is shared also by electrons in Bloch's states in a lattice. Is that a coincidence or is there a reason ? I mean, electron in a lattice should a potential which is not invariant under an arbitrary translation, so I find it a bit strange that the density of states is the same.

Thanks Ric

 

[vanhees71]

It's not coincidence but due to the involved Fourier transformations. For Bloch states $V$ is the volume of the Brillouin zone. Just count momentum states in a momentum box $\mathrm{d}^3 p$. You get $\mathrm{d}^3 p \frac{V}{(2 \pi \hbar)^3}$, and usually in QM you set $\hbar=1$.

 

[dRic2]

Thanks, I remember now. So essentially the reason why you get always the same result is that, as long as the electron (particles) are constrained in a finite portion of space you could expand their wave functions with a Fourier series (or transform) ?

 

[vanhees71]

In this case you have to impose boundary conditions. Usually one uses two kinds of boundary conditions:

(a) rigid boundary conditions: Here you assume that the particle is strictly confined in a finite region. E.g., you can choose a cuboid, cube, or sphere for that. That's a somewhat artificial way to describe a particle in a trap, and you get standing waves as solutions for the energy eigenmodes.

(b) periodic boundary conditions: That's the right thing to regularize the motion of particles in free space, e.g., in quantum field theory to get rid of some obstacles of the infinite-volume limit (which can be pretty subtle, as e.g., in relativistic QFT having to do with Haag's theorem, or the definition of the square of S-matrix elements though they have the energy-momentum conserving $\delta$ distribution, which cannot be squared without giving the operation some proper meaning). The advantage of periodic boundary conditions over that of rigid boundary conditions in this case is that you have a well-defined momentum operator as a self-adjoint operator (with the possible momentum eigenvalues discrete according to the periodicity conditions). Here you get of course moving waves (in the here modeled torus).

(c) Bloch states: Here you have some discrete space-symmetry group describing the periodic structure (of an idealized infinitely extended) crystal without any perturbations of its lattice. Mathematically it's quite similar to case (b).

 

[dRic2]

But in cases (b) and (c), although you have the same bc, a free electron has a different Hamiltonian that an electron in a lattice test you get the same density of states.