Number of possible wavefunctions only countably infinite?

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    Infinite Wavefunctions
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Discussion Overview

The discussion revolves around the nature of wavefunctions in quantum mechanics, specifically addressing whether the number of possible wavefunctions is countably infinite or uncountable. It also touches on the implications of black hole entropy and its relationship to physical properties. The scope includes theoretical considerations and mathematical reasoning.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether quantum numbers are always countable, noting that an electron can have any arbitrary momentum, suggesting a continuum of states.
  • One participant asserts that while quantum numbers may be countable, the admissible wavefunctions can be expanded into sums, implying that the set of all wavefunctions is uncountable.
  • Another participant mentions that probability amplitude functions are of integrable square, leading to a claim that this set is countable, though this is challenged by others.
  • A participant introduces the concept of Rigged Hilbert Spaces, arguing that these spaces can accommodate non-square integrable functions, which complicates the countability of wavefunctions.
  • There is a discussion about the nature of black hole entropy and its encoding of physical properties, with some participants expressing uncertainty about how entropy can represent different combinations of these properties.
  • One participant expresses that their understanding of quantum mechanics is self-taught, indicating a personal journey through various texts and the challenges faced in grasping these concepts.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement regarding the countability of wavefunctions and the nature of quantum states. Some assert that there are uncountably many wavefunctions, while others maintain that certain sets are countable. The discussion on black hole entropy also reveals differing views on its implications and encoding.

Contextual Notes

Participants highlight limitations in their understanding of quantum mechanics and the mathematical frameworks involved, such as the distinction between Hilbert spaces and Rigged Hilbert Spaces. There are unresolved questions about the implications of these mathematical structures on the nature of wavefunctions and entropy.

Who May Find This Useful

This discussion may be of interest to individuals exploring quantum mechanics, mathematical physics, and the theoretical underpinnings of black hole physics, particularly those with a background in mathematics seeking to understand the complexities of these topics.

  • #31
physwizard said:
this is not true for all cases. for eg., the free particle energy eigenfunctions form a continuous spectrum.
e^{i\frac{\sqrt{2mE}}{\hbar}x}


These functions are not modulus square integrable. His integration in ℝ is ∞
 
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  • #32
Physwizard, your wrote:
I see. As per google, rigged has two meanings:
rigged past participle, past tense of rig (Verb)
Verb
1) Make (a sailing ship or boat) ready for sailing by providing it with sails and rigging.
2) Manage or conduct (something) fraudulently so as to produce an advantageous result.
It appears you are interpreting it as per the first meaning, while I was interpreting as per the second which was why I took objection to it.
I do believe mathematical and physical quantities ought to be properly named.

It was named correctly, without ambiguity, in the original Russian (оснащенное гильбертово пространство). Here, this refers to (1), and not to (2). The problem was not in the naming, it is in the problems of translation.
 
  • #33
nomadreid said:
Physwizard, your wrote:It was named correctly, without ambiguity, in the original Russian (оснащенное гильбертово пространство). Here, this refers to (1), and not to (2). The problem was not in the naming, it is in the problems of translation.

oh. didn't know there was a russian angle!
 

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