Periodic boundary condition (Born-Von Karman): does the size of the cube matter? Periodic boundary conditions are chosen to faithfully represent the dynamical situation of electrons flowing through the metal. It's a better picture than the stationary wave one. We imagine a cube of size L and impose that the free electron wavefunction satisfies: $$\psi(\vec{r}+L\hat{i}+L\hat{j}+L\hat{j})=\psi(\vec{r})$$Sommerfeld theory uses this and leads to results that are independent of the chosen cube of side L. But here are my doubts:

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*The chosen L influences the allowed momenta. Also momenta are observables! Shouldn't observables be independent of our choice of L?

*Given a cube of metal, in Sommerfeld model do we schematize it as just one big cube of the same size and derive the allowed momenta? But why don't we use stationary conditions then? If we schematize it with a smaller cube, what should its size be? If we schematize it as a bunch of cubes, what should their size be? Is the lattice model of crystals related to this?

 A: 1.

The chosen L influences the allowed momenta. Also momenta are observables! Shouldn't observables be independent of our choice of L?

Yes, the chosen $L$ influences the momentum eigenstates which are obtained from periodic boundary conditions. However, these momentum eigenstates are not the real eigenstates of the electrons.
The point of the boundary conditions is not to work out the physical states of the electrons but to count how many states there are. For a large system, we make the continuum approximation, i.e. we assume that the spacing between consecutive momenta is so small that it is practically a continuous variable such that the dependence of the momentum on $L$ is no longer important. Furthermore, we are normally interested in quantities per unit volume so the factor $V=L^3$ drops out, which is why the results are independent of $L$.
If you could actually measure the momentum of a single electron, its state would collapse into one of the allowed momentum eigenstates. However, these would not be the ones derived from periodic boundary conditions but from the actual boundary conditions of the system. Actual boundary conditions are generally only taken into account for systems where quantum confinement is important e.g. quantum dots (nanometre sized semiconductors which confine the electrons in all three dimensions) that can be modelled as a (finite) potential well. The results definitely $do$ depend on $L$ in these cases.

2.

Given a cube of metal, in Sommerfeld model do we schematize it as just one big cube of the same size and derive the allowed momenta?

The quantization volume is arbitrary and nothing to do with the physical sides of the metal. You can choose it to correspond to the actual metal but it is not necessary. All that matters is that it is big enough to make the continuum approximation.

But why don't we use stationary conditions then?

Periodic boundary conditions yield complex exponential functions which 'happen to be' momentum eigenfunctions. They are a mathematical trick to give convenient functions to work with. Stationary wave boundary conditions on the other hand yield sine functions which are not momentum eigenfunctions. The introduction of a fake confining potential to achieve this is just another mathematical trick to fix the form of the functions. There are definitely not physical boundaries and no one is claiming that the walls of the metal approximate them. Indeed, as I said, the box doesn't even have to correspond to the sides of the metal.
The key point is that (focusing on one dimension for simplicity) any function $f(x)$ can be expanded on the interval $0<x<L$ as a Fourier series, but this can be either a complex exponential series or a sine series (or even a cosine series). The different types of boundary conditions are just ways of 'deriving' these different Fourier basis functions, any of which works equally well. Thus your statement

Periodic boundary conditions are chosen to faithfully represent the dynamical situation of electrons flowing through the metal. It's a better picture than the stationary wave one.

is not really correct as all boundary conditions are basically just a trick to obtain a certain form of basis function. The reason periodic boundary conditions are more popular is that momentum eigenstates are more convenient, e.g. we consider things like an electron with momentum $\boldsymbol{p}$ scattering into momentum $\boldsymbol{q}$.

If we schematize it with a smaller cube, what should its size be? If we schematize it as a bunch of cubes, what should their size be? Is the lattice model of crystals related to this?

As I said before, the size doesn't matter as long as it is large enough that we can take the continuum approximation and divide by the volume. I don't know why you would schematize it as a bunch of cubes and it is definitely not related to the lattice model of crystals: these boundary condition considerations apply in other cases such as particle physics and modes of an electromagnetic field.

As a final note, in Sommerfeld theory, the states obtained from periodic boundary conditions (momentum eigenstates) confusingly happen to be the actual energy eigenstates of the free electrons (in the continuum approximation). In general, the Hamiltonian and momentum operator do not commute, so eigenstates of momentum are not automatically eigenstates of energy. This is the case in a real metal, where electrons interact with the lattice resulting in Bloch states, superpositions of momentum states.
