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."(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]

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

[/tex]

This admits solutions of the form [itex]\psi(x) = Ae^{ikx} + Be^{-ikx}[/itex], where of course [itex]k = \sqrt{2mE}/\hbar[/itex]. 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?

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# Scattering states confuse me.

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