Section A3 of Chapter II on mathematical basis of QM in Cohen-Tannoudji’s "Quantum Mechanics" book has me a bit confused. The section is named Introduction of “bases” not belonging to F, and it comes right after the section explaining continuous orthonormal bases of state space. Let me copy a few fragments:
The $\{u_i(\mathbf{r})\}$ bases studied above are composed of square-integrable functions. It can also be convenient to introduce "bases" of functions not belonging to either $\mathcal{F}$ [this is the wave function space] or $L^2$ [this is the wider definition of Hilbert space], but in terms of which any wave function $\psi(\mathbf{r})$ can nevertheless be expanded. We are going to give examples of such bases and we shall show how it is possible to extend to them the important formulas established in the preceding section.
It then goes on to explain plane waves ($v_\mathbf{p}(\mathbf{r})$) and delta functions ($\xi_{\mathbf{r}_0}(\mathbf{r})$). At the end, it states:
The usefulness of the continuous bases that we have just introduced is revealed more clearly in what follows. However, we must not lose sight of the following point: a physical state must always correspond to a square-integrable wave function. In no case can $v_\mathbf{p}(\mathbf{r})$ or $v_{\mathbf{r}_0}(\mathbf{r})$ represent the state of a particle. These functions are nothing more than intermediaries, very useful in calculations involving operations on the wave functions $\psi(\mathbf{r})$ which are used to describe a physical state.
So, my first confusion was that it seems as though it is saying that we can expand functions in $\mathcal{F}$ in terms of a basis that is not in $\mathcal{F}$. How is this possible?
It talks also talks in the same section about the Fourier transform of $\psi$ and the relation with plane waves. As I understand (correct me if I'm wrong), one can switch between the position and momentum representations of the wave function by means of the Fourier transform. Reading the comments in another question, I found this comment, which seems to suggest that there exist some things called momentum wave functions, that arise from the canonical relation between $x$ and $p$ (not quite sure what this is either). Are those momentum wave functions referring to plane waves?
Also, I remember in a basic quantum mechanics course I did a few years back, it was mentioned that a plane wave could not determine the state of a system, but a wave packet, which is a collection of plane waves, could be a valid state. Does this have something to do with this discussion? How is it that a single plane wave is not possible (is not in $\mathcal{F}$) but a linear combination of them is?
