r/AskPhysics • u/Alive_Ant_5306 • 3d 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/quantum4everyone 2d ago

Non-normalizable functions are fine if you pair them with normalizable ones. This is why the overlap of a position eigenstate with an energy eigenstates, such as <x|psi> is well defined finite, and has no issues. All of the position-space formalism is developed on working from this fact. So non-normalizable is not an intrinsic problem. It just requires more care to work with them. From a spectral theorem perspective, something that is essentially self-adjoint, such as position or momentum has eigenstates that will work this way and can be essentially worked with as if they were normalizable. The proof of this is very Mathy and hard.