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Without going into all the equations, the Bohr model of the hydrogen atom uses
the assumptions that:
  • the light electron circulates around the much heavier proton, prevented from
    flying into space by the attractive electric force between the positively charged
    proton and the negatively charged electron;
  • the electron, although often acting like a particle, is actually a wave
    (some will say it has both particle and wave natures, but it's really a wave,
    and often the wave is so localized that it appears to be a particle), and it's
    wavelength is h/p, where h is Planck's constant, and p is the electron's
    momentum (so the faster it goes, the shorter the wavelength);
  • the electron fits in a circular orbit around the proton (this is a very
    simplified model), and the electron's wave cycles around the circle adding to
    itself with each cycle.
What we have pictured above is a proton in the center of a circle which represents
where the electron wave will reside. To get a feeling for the animation click the
"Decrease λ" button until the wavelength = C/3.0
(C = the circumference of the circle, and strictly speaking the circumference and radius
of the circle would change as the wavelength (and momentum) changes, but that would be
a technical pain to animate, so just imagine the scale of the whole thing changes instead).
Press the "Graph" button. What you should see is a nice wave with 3 peaks and 3 troughs
fitting perfectly around the circle. Now press the "Decrease λ" button
again, so the wavelength is now C/3.1. Press the "Graph" button. This time you get
3 peaks and troughs, but the start of the wave doesn't exactly match the end of the wave -
it's not a perfect fit.

Because it's a perfect fit in the wavelength = C/3.0 case (a perfect integral number of
waves around the circle), if I went around again and added the resulting wave to the first
wave and took the average it would look exactly the same. But if I did this for the
wavelength = C/3.1 case, the resulting average would be different than the once around
version. And each time I went around and added on an extra cycle that was a little
out of phase of the cycle before (and take the average of the total), the resulting
average would look more and more different.

That's what the "More Turns" and "Fewer Turns" buttons do. They allow the wave to
take from 1 to 10 cycles around, sum up all the cycles, and then find the average
and plot it. What you should find is the following:
  • If the wavelength = C/integer, then there is no change as you add
    more and more turns (the average of n identical things is just the thing);
  • If the wavelength is not C/integer, then the more turns (cycles) around
    you include, the closer the wave will fit onto the circle (and a perfect fit
    on the circle means NO WAVE, so NO ELECTRON - it interferes itself into
By the time you've included 10 turns around (the maximum I allowed), most waves
that do not fit perfectly around the circumference an integral number of times
will have pretty much interfered themselve to nothing (tight fit on circle).
Try it with wavelength = C/3.1; start with 1 cycle around, press "Graph"; go to
2 cycles around (use the "More Turns" button) and press "Graph"; do this until you've
got 10 cycles around. By that time the wave should be pretty much gone. This idea
was used to visualize why electron energy levels are discrete.