A question about wave function

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Discussion Overview

The discussion revolves around the properties of wave functions in quantum mechanics, specifically addressing why a wave function must approach zero at infinity. Participants explore the implications of this requirement for different types of states, including bound and scattering states, and the normalization of wave functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of a wave function being zero at infinity, suggesting that particles could be found throughout the universe.
  • Another participant explains that bound states have energies less than the potential energy at infinity, leading to wave functions that approach zero at infinity, while scattering states can have wave functions that do not go to zero at infinity.
  • There is a discussion about the normalization of wave functions, with a participant emphasizing that the total probability must equal 1, which implies that wave functions must approach zero at infinity for certain states.
  • One participant argues that if a wave function does not go to zero at infinity, it would imply a surprising level of quantum delocalization over large distances.
  • Another participant notes that for free particles, the wave function is constant and not normalizable, leading to a uniform probability distribution across space.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of wave functions approaching zero at infinity, with some supporting the idea based on normalization and physical assumptions, while others challenge it by considering the nature of scattering states and free particles.

Contextual Notes

The discussion highlights the dependence on definitions of bound and scattering states, as well as the implications of normalization on the interpretation of wave functions. There are unresolved questions regarding the nature of particles in relation to their wave functions at infinity.

coki2000
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Hello physicists,

My question is why a wave function has to be 0 at infinitiy. The author who I was studying him quantum book explains my question by saying that it is necessary to make wave function normalizable. But in my opinion there can be an arbitrary particle which can be found all around the universe therefore at infinity. Please illuminate me:smile: thank you all
 
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There are two kinds of states for wave functions: bound states and scattering states. For bound states the energy of the particle is less than the potential energy at infinity, such a particle cannot be found at infinity and the wave function goes to zero at infinity. On the other hand for scattering states the energy of the particle is more than the potential energy at infinity. In this case the particle can be found anywhere and the wave function does not go to zero at infinity. The wave function for a free particle is such an example. The wave function in this case is not normalizable.
 
Thank you Avijeet,
But for a particle, the probability to stand anywhere in space must be 1, mustn't it?
 
You should think of it as a physically reasonable assumption you choose to make. If the wavefunction were not zero at infinity, wouldn't you find that surprising? :) We don't tend to encounter quantum delocalization over very large distances. Hence, this assumption.

Also: consider the normalized wavefunction (or its modulus-squared, the probability density). There is only so much probability to go around: it all has to add to 1. If your probability does not go to zero at infinity, the particle will be so "spread out" that you will never detect it in your laboratory -- so why are you analyzing that particle, instead of one that's a little closer to home?
 
coki2000 said:
Thank you Avijeet,
But for a particle, the probability to stand anywhere in space must be 1, mustn't it?

For a free particle, the amplitude of the wave function at any point is a constant. The wave function is not normalizable, so the probability interpretation of the wave function is not applicable. All you can say is that the particle is equally probable at all the points.
 

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