Wavefunction normalization vs 1/√(|x| )

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Discussion Overview

The discussion revolves around the normalization condition for wavefunctions in quantum mechanics, specifically addressing the requirement that a wavefunction must approach zero faster than 1/√(|x|) as x tends to infinity. Participants explore the derivation and implications of this condition, focusing on the concept of square integrability and its relation to physical interpretations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant cites Griffiths' text, stating that a wavefunction must approach zero faster than 1/√(|x|) for normalization.
  • Another participant explains that the wavefunction must be square integrable, implying it must fall off to zero faster than certain non-normalizable functions.
  • A different viewpoint suggests that there are square integrable functions that do not necessarily approach zero at infinity, indicating a broader interpretation of the condition.
  • One participant acknowledges the need for the wavefunction to be square integrable while also satisfying other physical conditions, emphasizing the importance of the probability interpretation.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the normalization condition, with some agreeing on the necessity of square integrability while others challenge the sufficiency of the condition alone. The discussion remains unresolved regarding the broader implications of these interpretations.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the nature of wavefunctions and the specific definitions of square integrability. The relationship between physical conditions and mathematical requirements is also not fully explored.

Maharshi Roy
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In Griffith quantum mechanics, it is written that for a wavefunction to be normalizable, it is essential that the wavefunction approaches zero before 1/ √(|x|) as x tends to infinity...
Please explain from where this condition has been derived.
 
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The idea is that the wave function must be square integrable, so it must fall off to zero faster than functions which aren't.

Or put another way, integrate \left( 1/\sqrt{|x|}\right)^2 and see what you get.
 
There are square integrable functions which don't fall off to 0 when going to infinity. Griffiths' condition nicely singles out the so-called Schwartz test functions.
 
You're right. I guess I should have said that the wave function must be square integrable and satisfy all other physical conditions of wave functions. (It certainly would be weird if the probability of finding the particle infinitely far away is non-zero.)
 

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