Pψ=aψ and wave function uniqueness

[mengsk]

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!

 

[CompuChip]

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?

 

[OOO]

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.

 

[mengsk]

I mean can we find a ψ to make the equation tenable?

 

[OOO]

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.

 

[mengsk]

Thank you very much!