What makes localized energy eigenstates, localized?

Click For Summary

Discussion Overview

The discussion revolves around the concept of localized energy eigenstates in quantum mechanics, particularly in the context of the time-independent Schrödinger equation. Participants explore the meaning of localization, its implications for probability density, and the distinction between localized and non-localized states.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the term "localized" in relation to energy eigenstates, suggesting it may pertain to whether the eigenstate is bounded.
  • Another participant explains that localization informally refers to the probability density being confined to a region of space much smaller than the entire system, implying finite variance in position.
  • A participant questions how to demonstrate the non-localization of a plane wave, expressing confusion over the informal use of "most" regarding probability density.
  • It is noted that a plane wave is not square integrable, thus not a legitimate quantum state, and is inherently non-localizable.
  • A reference to Rigged Hilbert Spaces is made, suggesting that understanding distribution theory could enhance comprehension of these concepts.

Areas of Agreement / Disagreement

Participants express differing views on the definition and implications of localization, particularly regarding the nature of non-localized states like plane waves. The discussion remains unresolved with multiple competing interpretations of localization.

Contextual Notes

There are limitations in the discussion regarding the definitions of localization and the mathematical framework necessary to fully understand the implications of non-localized states. The reliance on informal language and assumptions about probability density adds to the complexity.

Zacarias Nason
Messages
67
Reaction score
4
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.
 
Last edited:
Physics news on Phys.org
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.
 
  • Like
Likes   Reactions: bhobba and Zacarias Nason
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.
 
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
 
Last edited by a moderator:
  • Like
Likes   Reactions: Zacarias Nason

Similar threads

Undergrad Symmetry of Hamiltonian and eigenstates
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Are All Solutions of the 1D Schrodinger Equation Energy Eigenstates?
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Physical interpretation of phase in solutions to Schrodinger's Eqn?
  • · Replies 12 ·
Replies
12
Views
5K
Graduate Confusion about 2016 Unruh paper - fluctuating vacuum energy density?
  • · Replies 27 ·
Replies
27
Views
4K
Graduate The Schrodinger Equation: How to Solve for Time Evolution and Eigenstates
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Time-dependent unitary transformations of the Hamiltonian
  • · Replies 9 ·
Replies
9
Views
7K
Graduate What is the connection between energy eigenstates and position?
  • · Replies 2 ·
Replies
2
Views
5K
High School QFT & QM Questions (about time, locality, gravity, etc.)
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Energy Eigenstates: Solving Schrodinger Equation & Time-Indep
  • · Replies 6 ·
Replies
6
Views
6K
Graduate TIme-independent schrodinger equation
  • · Replies 1 ·
Replies
1
Views
2K