Using a quantum microscope, it is possible to view the wave function of the hydrogen atom (see diagram above). Note the electron can only be found in the lit areas and not in the dark areas between. When the electron moves between energy levels, it seems to mysteriously vanish from our universe and then mysteriously reappears, hence the alternating pattern of dark and light circular bands.
Of course there's a string theory that explains this phenomenon: when the electron leaves an energy level it literally leaves our 3D space and enters a higher curled dimension. It loops around then re-enters our space:
At the diagram above you can imagine the electron (red dot ) moving left to right. The loop represents the extra hidden dimension.
Now notice the diagram below. It is the famous double-slit experiment. Notice the target screen has light and dark areas. We could postulate that the dark areas are due to the light disappearing into a higher dimension or an alternate universe--or perhaps it was abducted by extraterrestrials.
Of course this is all nonsense. We know why the dark sections are dark and why the light sections are light: constructive and destructive interference wave patterns. Where the red and black lines are parallel there's light (constructive interference), where they cross or move in opposite directions, there is darkness (destructive interference).
Since there is no empirical evidence of extra dimensions, it makes sense to propound an alternate hypothesis that uses established physics as its basis--the established physics being constructive and destructive interference. If you compare the two diagrams above, you will note a striking similarity: both have alternating light and dark areas on their respective target screens.
We know what causes the interference pattern of the double-slit experiment--it's the two slits. But what could possibly cause the interference pattern (discrete energy levels) of the hydrogen atom? That's what this post shall cover. First, let's define the variables needed:
At equation 1 below, energy (E) is charge (q) times voltage (V). Assuming energy E is an eigenvalue, it must have a probability (P(E)). That brings us to equation 2. Quantum mechanics tells us the probability is an amplitude squared (A^2, see equation 3). A pinch of algebra gives equations 5 and 6.
At equation 5 we see the positive charge from the hydrogen atom's proton corresponds to a positive amplitude squared. At 6 the negative charge from the electron corresponds to a negative amplitude squared. To get a negative amplitude squared we multiply the positive amplitude of a sine wave with its negative amplitude:
Notice there's no way to get a positive amplitude squared using this method. To get the positive squared amplitude, we need to do the following:
We use the absolute value of sine to create a positive squared amplitude. We now have what we need to model the hydrogen atom's constructive and destructive interference wave pattern. Check out equations 7 through 10 below:
When we plug in values for equation 7, we get the wave pattern below which maps beautifully to the light and dark areas of the hydrogen atom:
No extra dimensions needed, just old-school physics.
