Is there a definition of a localized state in quantum mechanics? I've seen that some textbooks use this term, but they don't give a definition.
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A localized state of a particle or a field means the same thing as in semi-colloquial context: it means that most of the energy density (for the case of a field) or most of the probability (particle, quantum mechanics) may be found within a region (e.g. a ball) whose linear size is smaller (or much smaller) than the total size of the material (and surely smaller than infinity).
The opposite term is a delocalized state – e.g. a plane wave – in which the particle may be found anywhere in the material or anywhere in space.
answered Nov 10, 2011 at 11:52
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$\begingroup$ I suppose there's no "hard", precise definition? $\endgroup$– a06eCommented Sep 8, 2013 at 1:36
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$\begingroup$ The ‘hard’ definition, when you can make sense of it, is that a localized state is a vector in the Hilbert space. A delocalized state is not normalizable so it doesn’t belong to the Hilbert space. $\endgroup$– lcvCommented Nov 22, 2017 at 10:00
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Newton and Wigner had a more precise definition.
https://www.google.co.jp/search?q=wigner+newton&ie=UTF-8&oe=UTF-8&hl=en-jp&client=safari
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2$\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$– Community BotCommented Sep 29, 2022 at 13:36