# Charge distribution of the Hydrogen atom

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The time-averaged potential of a neutral hydrogen atom is given by 4s (1 + 7) wo r where q is the magnitude of the electronic charge, and cx-' = ao/2, ao being the Bohr radius. Find the distribution of charge (both continuous and discrete) that will give this potential and interpret your result physically.

It is taken from "Classical Electrodynamic 3rd" by Jackson and I know it will use the Poisson Equation.

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111Equation Chapter 1 Section 1

Lets rewrite the potential as:

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Poisson equation states that:

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Therefore:

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The Del operator in spherical coordinates is:

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The Laplacian in spherical coordinates is given by:

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We use the property of the Laplacian of a product:

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In our case:

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The functions are angular-independent. Therefore:

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And:

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Where we used the known relations (see appendix for proof)

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For the last term:

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So we get:

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We know that the delta function is zero everywhere except at the origin, where so with no loss of information we can write it as:

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The first term is the charge of a proton (discrete point charge) located at the origin.

The second term is the negative charge of an electron smeared over a cloud that decays exponentially.

Appendix: The Laplacian of 1/r

We would like to calculate the Laplacian of

According to (1.5) this gives:

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But this is true only if , otherwise we encounter singularities.

So let's integrate (Please see the attached file) over a volume of sphere of radius a :

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Using Stokes theorem:

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Where S is now the surface of the sphere.

Now:

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On the surface of the sphere

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And:

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Where [omega] is the solid angle.

Plugging this into (1.15) we get:

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Since the solid angle of an entire sphere is 4[pi]

So we have:

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It is zero outside of the origin, but if we integrate it over any sphere that contains the origin we get a finite value.

We recall that by definition, and for any for any volume that contains the origin:

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Therefore we can write (1.20) as:

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