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Very tiny question Which do you think is right?

Think a hydrogen atom floating in the vacuum. Suppose that its electronic state is 1S state.

Conventional analytical solution of 1S state represents a motion of a reduced mass around the center of mass between the electron and the proton.  The proton's mass is known to be about 1836 times heavier than that of the electron. In the hydrogen atom system the proton is wobbling like a star associated with an exoplanet.

So the quantum mechanical cloud of the electron, the lighter particle, spreads over much larger space than that of the proton, the 1836times heavier particle.  In other word, the wave function of the proton is very tightly compressed when compared with that of the electron. Mathematically speaking and expressing simple, their wave function may be expressed like



where r is the radius in suitable unit system and the mormalization constants are omitted.

Then think a virtual sphere of radius R centered on the hydrogen atom.  If we integrate the square of each wave function in this spherical space increasing the R to infinity, then we can obtain the full proton charge +q and the full electron charge -q.

Here comes the tiny question.

If the radius parameter R remains finite value, then the total net charge included in this sphere is,

a. Positive.  Because for proton, even the integration to a finite radius R results almost near the full proton charge +q, while the integration of the electron cannot reach to the full charge -q enough.

b. Neutral.  Because something is wrong in the above calculation.  And there has been ever no such a story that a hydrogen atom was observed to possess a finite residual charge which depends on the distance between the atom and the observing point.

Which do you think is right?

Or do you have any other story?  Please show it.


January 17, 2012

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