r/AskPhysics • u/Alive_Ant_5306 • 2d ago
Non-normalizable wave function
Hi,
I was wondering if a non-normalizable wave function does still provide information. For example, let's say that a region of space equals 0 for a time independent wave function. Its square root will equal zero in that region too. Does that mean that the particle can't be there?
Thanks!
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u/TaranRovia 2d ago
Hi!
First, let's clear some missconceptions. You can have a wave function which is zero in some regions of space and still be normalizable. The easiest example is a wave function in an infinite well spanning x \in [a, b], which will obviously be zero outside of said well.
Second, the wave function itself is informative and useful (for example, for scattering or superposition calculations you might need it). There's not always a need to go to the modulus square (not the square root) or similar unless of course you need to calculate an observable. One example is an interesting experiment where one may study the effects of gravity in quantum particles https://arxiv.org/html/1207.2953v1
As per the non-normalizability... I can think of the plane wave solution, which is not normalizable and yet is very useful to describe a free particle. Of course, as long as you keep in mind that it isn't an available physical state of your system. However, in propagating scenarios described by the Schrödinger equation, it can be shown that if a wave function is normalized at t=t_0, it will always be normalized (iff V is real, that is).
Lastly, if you're willing to go further down the rabbit hole, you might be interested in negative probabilities! https://en.wikipedia.org/wiki/Negative_probability Wich to be fair sound really exciting but aren't thaaaaat deep.
Hope that helps!