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Considering the double slit experiment with a charged particle, after the particle passes through the slits, do the two portions of the wavefunctions feel the electromagnetic attraction of the other portion?

More generally, can the potential term in the Schrödinger equation be dependant on the wavefunction itself?

On the one hand it is easy to imagine an electron wave splitting up, each part carrying a proportion of the charge. On the other hand, the wavefunction is a probability distribution, which implies that there is no interaction.

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    $\begingroup$ Wavefunctions are mathematical constructs that live on pieces of paper that were scribbled on by physicists. We usually do not believe that these symbols interact with themselves. Apologies, I couldn't resist the urge to point out the flaw of mistaking a human description of reality for reality. The correct answer to your question is: no, one can not construct generally useful models of reality outside of limited problem sets in molecular and solid state physics by using semi-classical mean field approximations of a non-relativistic wave function. For certain problems, however, it works. $\endgroup$
    – CuriousOne
    Aug 12, 2014 at 20:19

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"More generally, can the potential term in the Schrödinger equation be dependent on the wavefunction itself?"

The answer is negative: The resulting Schroedinger-like equation would turn out to be non-linear. It would not be associated with a unitary time evolutor (the self-adjoint generator, the Hamiltonian operator, would not be defined) against some basic postulate of quantum mechanics.

However non-linear Schroedinger-like equations where the potential depends on the wavefunction itself have great physical interest, but they describe systems of many particles. I am referring, in particular, to the so-called Gross-Pitaevskii equation.

In the double slit experiment, the "self-interaction" of the wavefunction is a linear phenomenon, the superposition principle applied to the two parts of the same wavefunction emitted by the two sources (the slits). Non-linearity arises as soon as you compute the probability amplitude just in view of the mathematical procedure.

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  • $\begingroup$ +1, the answer (at least part of which) i never gave :)) $\endgroup$
    – Nikos M.
    Aug 12, 2014 at 20:39

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