Does the semiclassical model contradict statistical mechanics?

Question

In the discussion of electrons in metals, often the semiclassical model is used. There, the electrons are treated as occupying localized wave packets $|k,x\rangle$ which have momentum k and position x with small fluctuations.

However, we know from statistical mechanics that the state of the system is given by the density operator

$$\rho = \frac{e^{-\beta H}}{Z} $$

As we know from the theory of mixed quantum states, this means the electron occupies an eigenstate of H, we just don't know which one it is. It seems reasonable to assume that the eigenfunctions of H are delocalized in space (*).

Question: Therefore, the semiclassical model is in direct conflict with statistical mechanics, isn't it?

$$\\$$ I'm especially concerned about conventional treatment of homogeneous semiconductors, or metals. For those systems, we use both the semiclassical model, aswell as the statistical mechanics of non-interacting fermions.

$$ \\ $$ (*) I guess this depends on the exact model we are using. If we ignore electron-electron and electron-phonon interactions, and include dilute impurities (doping atoms or lattice defects) it should be correct I think. The Bloch waves are still solutions of the stationary Schrödinger equation everywhere but at the points of the impurities. Additionally, if the eigenfunctions of H are really not delocalized in any actual system in nature (or put differently, if the eigenfunctions don't resemble Bloch functions at all), it seems the Bloch model and the band theory of metals must be inappropriate for actual systems.

Answer 1

I think we should speak here not of semiclassical model, but of semiclassical approximation, which preserves some of the quantum features in favor making things more tractable.

Indeed, if we were to demand that the position and momentum can be measured simulatenously, we could not have quantum effects. However, if we demand that they are measured simultaneously up to certain precisions, $\sigma_x,\sigma_p$, we would not run into much difficulties, as long as $\sigma_x\sigma_p \geq \frac{\hbar}{2}$.

Quantum kinetic equations
In practice this is usually achieved by writing up the fully quantum kinetic equations and then deriving approximations in terms of some small parameter, e.g., the ration between the electron wave lengths and the characteristic scale of the potential. Such derivations are presented in the review by Rammer&Smith and the book by Kadanoff and Baym (The generalized kinetic equations then often passes under the name of Kadanoff-Baym equations.) However, these derivatiosn require knwoledge of quantum field theoretical methods (as applied to statistical physics) - such a knowledge is usually not expected from the readers of the books on the theory of semiconductors and semiconductor devices, such as Ashkroft&Mermin, which therefore resort to confusing hand-waving explanations.

Wigner function
Returning to Kadanoff and Baym - these base their derivation on the use of Wigner function, which is the closest that we can get to having a consistent wave packet picture without destroying the quantum mechanics: $$ W(x, p)=\frac{1}{\pi\hbar}\int_{-\infty}^{+\infty}\psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}dy $$ Although this function is well defined quantum mechanically, its interpretation as a probability encounters difficulties, as it can take negative values. These can be neglected only in the context of consistent approximations, as mentioned above.

Answer 2

First of all, it's important to remember that we are dealing with models here, which are mathematical expressions that are derived from certain assumptions. Generally speaking, in materials systems we need to work with very large numbers of particles that can interact in non-trivial ways, so there is a tendency to simplify things to make the mathematics tractable.

In metals and semiconductors we have an electronic band structure which tends to fill the lowest energy states first, with some variation due to thermal energy. If we are to use a statistical mechanics approach there is a complication here: we need to quantify the H in your equation. What potential does our electron "see", and where do these quantized states lie? Statistical mechanics doesn't tell us how to do this: we need another approach.

The semiclassical model is an attempt to address this question by treating the potential within a solid as a small, periodic perturbation. This allows us to approximate the band structure of a solid, which then forms the basis for subsequent calculations of electron transport and so on. This is an approximate approach, and it need not converge with statistical mechanics at all.

Hypothetically, if you knew the electron energy levels and all possible microstates precisely you could apply statistical dynamics and compare that to the numbers you get from the semiclassical approach. You will also get different numbers. But you simply will not get an analytical solution for this approach, so you'll need to run some pretty heavy numerical approximations to do it.

So in summary: yes there is a contradiction (the semiclassical model makes some simplifying approximations statistical mechanics does not), but it may be tolerable in contexts where the semiclassical model is a good approximation and you don't have access to a supercomputer or the specific codes for your specific material system. As with all things in materials science, whether an approximation is good or not is determined by experiment and real-world usefulness, not theory, so non-convergence with a more fundamental theory isn't the problem you may think it is.

On the other hand, if you want to know why the semiclassical model works as well as it does, all I can really say is that the error caused by not using a true quantum treatment must be small in the contexts where it is used.