The interpretation of Schrodinger's wave function for a nonrelativistic charged particle is that the wave function square represents a position probability density of that particle based on Max Born's interpretation. On the basis of the Klein-Gordon equation, the wave function square is interpreted as charge density, while according to Dirac equation, the interpretation of wave function square is the same Schrodinger's theory.
You may also be interested in Schrödinger's own forgotten opinion on the wave function. Brought to the forefront again with this recent paper titled "The Hydrogen Atom Fundamental Resonance States":
https://file.scirp.org/pdf/JMP_2018042716061246.pdf
I think Schrodinger's own opinion was disregard, Max Born' interpretation has been adopted formally. Is not it prof André Michaud.
Dear Ali,
I agree. This is what happened, unfortunately for the progress of physics, because Schrödinger and de Broglie did not exclude the possibility that the electron might remain localized even as it was axially resonating within the volume defined by the wave equation..
Dear Ali Chassib, on how de Broglie introduced the wave function to end up to its particle-wave duality relation, you may also refer to this article
https://reader.elsevier.com/reader/sd/pii/S1631070517300774?token=8366D9436CF18C3F3FE58DD16BA3CEA8F3A21541B87EDF3B5323DE11FF39F90789E76FDC6CE00D0FBCB6B45A6867D00E
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