Why are plane waves not possible representations of states in quantum theory?

[AxiomOfChoice]

One of the postulates of quantum mechanics, as quoted in any textbook on the subject, is something like the following: "states are vectors in a Hilbert space."

But then they go on to solve the problem of the free particle, which should (I guess) be about the simplest problem one can solve. The associated stationary Schrödinger equation in one dimension looks like

$$-\frac{\hbar^2}{2m} \psi'' = E\psi.$$

This admits solutions of the form $\psi(x) = Ae^{ikx} + Be^{-ikx}$, where of course $k = \sqrt{2mE}/\hbar$. But these are NOT normalizable and are therefore not in any sort of Hilbert space, since any vector in a Hilbert space necessarily has finite norm (and can therefore be normalized). So what is going on here?

 

[jtbell]

Those states are not physically realizable. Physically realizable scattering states are superpositions of them (wave packets). Nevertheless, they are useful as idealizations.

 

[AxiomOfChoice]

Those states are not physically realizable. Physically realizable scattering states are superpositions of them (wave packets). Nevertheless, they are useful as idealizations.

Okay. So though those states themselves are not in our Hilbert space, certain linear combinations (and I guess we need to talk about $\int$ instead of $\sum$ when we say "linear combinations"...or do we?) of them are?

 

[vanhees71]

Plane waves are not possible representations of states in quantum theory. They are distributions and belong to the dual of the domain of the position and momentum operators and all their powers. This domain is a dense subspace of the Hilbert space, and it's dual is thus much larger than the Hilbert space (for a Hilbert space the topological dual is isomorphic with the Hilbert space itself).

This formulation of quantum theory, called Gelfand construction (or rigged Hilbert space), justifies the quite handwaving approach of physicists to these matters, which goes back to Dirac. A very nice pedagogical introduction can be found in

R. de la Madrid, The role of rigged Hilbert space in quantum mechanics, Eur. J. Phys. 26, 287 (2005) 287
doi:10.1088/0143-0807/26/2/008

A good textbook using this formulation is

L. Ballentine, Quantum Mechanics

and a more formal mathematical representation is given in

Galindo, A., and Pascual, P.: Quantum Mechanics, Springer Verlag, 1990, 2 Vols.