What makes localized energy eigenstates, localized?

In summary, "localization" in the context of quantum mechanics refers to the property of an energy eigenstate being confined to a region of space significantly smaller than the entire system. This results in a finite variance in position and is typically seen in localized quantum states. However, a plane wave is an example of a non-localized quantum state and is not considered a legitimate quantum state because it is not square integrable. The concept of localization is handled in quantum mechanics through the use of a Rigged Hilbert space, which involves studying distribution theory.
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Zacarias Nason
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I'm reading about stationary states in QM and the following line, when discussing the time-independent, one-dimensional, non-relativist Schrodinger 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.
 
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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.
 
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  • #3
MisterX said:
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.
 
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Zacarias Nason said:
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
 
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Related to What makes localized energy eigenstates, localized?

1. What do we mean by "localized energy eigenstates"?

Localized energy eigenstates refer to the specific energy states that a particle or system can occupy within a given region of space. These states are characterized by a definite energy value and are confined to a specific region, unlike unlocalized states which are spread out over a larger area.

2. How are localized energy eigenstates different from non-localized states?

Localized energy eigenstates have a well-defined energy value and are confined to a specific region, while non-localized states have a range of possible energy values and can exist in multiple regions of space at the same time. In other words, localized states are more specific and well-defined, while non-localized states are more spread out and uncertain.

3. What causes a state to become localized?

The localization of a state is caused by the interaction of the particle or system with its surroundings. This interaction creates a potential energy well that confines the particle or system to a specific region of space, leading to the formation of localized energy eigenstates. In other words, the potential energy landscape created by the surroundings determines the localization of the state.

4. Can localized energy eigenstates exist in all types of systems?

Yes, localized energy eigenstates can exist in all types of systems, including atoms, molecules, and larger systems. However, the degree of localization may vary depending on the specific system and its surroundings. In some systems, the states may be highly localized, while in others they may be partially localized.

5. How are localized energy eigenstates important in quantum mechanics?

Localized energy eigenstates are crucial in quantum mechanics as they provide a way to describe the energy states of a particle or system in a specific region of space. These states play a crucial role in determining the behavior and properties of quantum systems, and understanding them is essential for studying and predicting the behavior of matter at the atomic and subatomic level.

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