quantum mechanics - Paradox of the wavefunction collapse into un unphysical state

"A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement."

— P.A.M. Dirac, The Principles of Quantum Mechanics

This is one of the postulates of quantum mechanics. However, there are some cases in which this statement leads to contradictions.

For example, we know that the eigenfunctions of the momentum operator (in 1D for simplicity)

$$\hat p = -i \hbar \frac{\partial}{\partial x}$$

are plane waves:

$$\psi_p(x) = A e^{ipx/\hbar}$$

These eigenfunctions are not normalizable and therefore are not acceptable as physical states.

If we try to apply the cited postulate to the momentum operator, we would therefore incur in a contradiction: the system cannot jump into an eigenstate of the momentum operator, because such an eigenstate would not be normalizable and therefore would not be a physical state.

This paradox is usually dismissed by saying that this line of reasoning applies to an ideal measurement, which cannot be realized in practice, and that for non-ideal measurement the situation is different. But this answer doesn't seem to be satisfying to me: although it makes sense, it is not clear what is the theoretical reason why an ideal measurement is not realizable.

There seem to be only two possible solutions to this paradox:

1. The cited postulate is wrong.


2. The momentum operator is somewhat ill-defined: for example, maybe we cannot just take its domain to be the set of all sufficiently regular (*) functions $f \in L^2(\mathbb R)$ as we usually do. In this case, maybe it is possible to give a definition of the momentum operator which agrees with the cited postulate.

What is a possible solution to this paradox?

PS: As far as I'm concerned, it is perfectly fine to answer that the solution is that an ideal measurement is not physically realizable in practice, but only if such a claim is backed up with rigorous theoretical arguments explaining why this is the case.

(*) Sometimes, the condition imposed is the absolute continuity of $f$, but I don't know if it can be relaxed.

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Updates

• Related questions and answers:

-Measurement of observables with continuous spectrum: State of the system afterwards (suggested by ACuriousMind). After some discussion, the author added a wonderful Addendum that maybe can be considered as an answer to this question.

-Quantum mechanics - measuring position.

• Related articles:

I found this article and this article (free download) which are about this exact problem, but they are quite technical and I still have to properly dig into them.