Problem with momentum operator

Question

Why is there no problem with the eigenfunction of the momentum operator being non-normalisable? How can it be a valid quantum state?

Answer 1

It is not a valid quantum state, it is an idealization of very long wave-packets emitted by atom-lasers. These wave-packets are almost coherent waves, very close, by their quantum description, to Fourier components, though they have finite length, e.g. 0.35mm (see arXiv quant-ph/9812.258, "An Atom Laser with a cw Output Coupler", by Bloch-Hänsch-Esslinger, and see the picture below).

As to the actual Fourier components, they are normalizable to Dirac $\delta$ function, s.t. we can work with them, $\int_{-\infty}^{+\infty} e^{ik'x}. e^{-ikx} dx = \delta(k - k')$.

Atom laser output: A collimated atomic beam derived from a Bose-Einstein condensate . In the upper side of the figure one can see the condensate, typically $^{87}$Rb.