Pψ=aψ and wave function uniqueness

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Discussion Overview

The discussion centers on the uniqueness of the wave function for a particle and its relation to the equation Pψ=Aψ, where P represents the momentum operator and A is a constant. Participants explore the implications of this equation in the context of quantum mechanics, including the conditions under which a wave function can be considered unique or non-unique.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants question whether the wave function of a particle is unique, noting that it is subject to initial conditions and can be transformed through gauge transformations.
  • One participant suggests that if P is treated as a first-order partial differential operator, the solution is unique up to a constant multiplicative factor, which can be used for normalization.
  • Another participant seeks clarification on the term "rationalize," indicating a desire to understand if a suitable wave function can be found to satisfy the equation Pψ=Aψ.
  • It is proposed that a plane wave can satisfy the equation, but this is not representative of the general wave function for particles that interact.

Areas of Agreement / Disagreement

Participants express differing views on the uniqueness of the wave function, with some asserting that it is not unique due to gauge transformations and initial conditions, while others suggest that solutions can be unique under certain mathematical conditions. The discussion remains unresolved regarding the broader implications of these points.

Contextual Notes

Limitations include the dependence on definitions of uniqueness and the specific conditions under which the momentum operator is applied. The discussion does not resolve the mathematical steps involved in determining the uniqueness of wave functions.

mengsk
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I want to know whether the wave function of particle is unique? If not, could we find a ψ to rationalize the equation Pψ=Aψ, in which P is the momentum operator and A is a constant. Thank you!
 
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In case P is just an ordinary (in this case, first-order partial) differential operator, all the ordinary results for existence and uniqueness of a solution hold.
AFAIK, for the momentum operator the solution is unique (of course, up to a constant multiplicative factor, which can be used to normalize the solution).
Not sure what you mean by rationalize though?
 
mengsk said:
I want to know whether the wave function of particle is unique? If not, could we find a ψ to rationalize the equation Pψ=Aψ, in which P is the momentum operator and A is a constant. Thank you!

What do you mean by "unique" ? The wave function is, of course, subject to initial conditions and thus not unique in the usual sense. Nor is it unique when IC are completely specified, because it can still be gauge transformed.
 
I mean can we find a ψ to make the equation tenable?
 
mengsk said:
I mean can we find a ψ to make the equation tenable?

Yes, in the case you mentioned Psi is simply a plane wave. But this is not the general wave function of a particle. Particles interact and then their wave functions are not plane waves anymore.
 
Thank you very much!
 

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