What makes localized energy eigenstates, localized?

[Zacarias Nason]

I'm reading about stationary states in QM and the following line, when discussing the time-independent, one-dimensional, non-relativist Schrödinger eqn, normalization or the lack thereof, and the Hamiltonian, this is mentioned:

"In the spectrum of a Hamiltonian, localized energy eigenstates are particularly important."

After that, the word "localized" is never apparently used again and hasn't been used prior in the text (these are lecture notes). What does localization mean here? If this is related to linear algebra, I haven't taken it and consequently don't know what it means. I'm having to assume for the current time that localization has to do with whether the energy eigenstate is bounded or not, but I'm not sure.

 

[MisterX]

Informally it means most of the probability density is confined in a region of space which is significantly smaller than the entire system. We would expect finite variance in position from this. An example of a non-localized quantum state would be a plane wave.

 

[Zacarias Nason]

An example of a non-localized quantum state would be a plane wave.

And how would you show this for a plane wave? I'd imagine it would involve calculating the probability density of a plane wave, but I'm thrown off by the informal "most" of the probability part. That sounds like setting some arbitrary threshold wherein in some small region some percentage of the probability is in it, but that doesn't sound right.

 

[bhobba]

And how would you show this for a plane wave?

Since a plane wave is not square integrable its not a legit quantum state. If it was then it would exist throughout all space so is obviously not localizable.

The way its handled in QM is you are really dealing with a Rigged Hilbert Space:
https://en.wikipedia.org/wiki/Rigged_Hilbert_space

You will understand it a lot better if you study distribution theory which IMHO should be in the tool-kit of any applied mathematician, not just those into physics
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

Thanks
Bill