Do Electrons Leave Our Universe When They Move Inside the Atom?

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.

Conquering the Infinite Slit Experiment

There once was a physics student named Richard Feynman who asked his professor (re: the double-slit experiment), "What happens if you increase the number of slits to infinity?" This question led to summing the infinite number of paths a particle can take from point A to point B. If each path has a finite energy and we add up all those energies, we should get infinite energy! But that can't be right, since we started with one particle with a finite energy. Energy should be conserved, so what's wrong here? That's what we shall address in this post. First, let's define the variables we will use:

To tame this infinity problem we borrow a great idea from calculus: the integral. The integral is used to find the area beneath a curve (see diagram below). Equations 1 and 2 define the integral in terms of a summation.

At equation 2, notice how we can take N (the number of y's) to infinity and end up with a finite area beneath the curve. A finite number is definitely what we are after. Another way to get the finite area under the curve is to determine the average y (see equation 3), then map our curve to a simple rectangle (see diagram below) that has the same area as the curve shape. Equation 4 shows that the integral is equal to 'a' (the x value) times the average y value.

Now, let's apply what we know to the slit experiment. At the bottom of the next diagram, a gun fires an electron with energy Ee. Let's assume that all lights and detectors are off and the electron goes through all the slits simultaneously. The energy at each slit is some multiple or fraction of Ee. These energies can vary due to constructive and destructive interference. As with our y's above, we determine the average epsilon or coefficient for each Ee.

We don't know the average energy per slit. We do know it must be greater than zero according to the Heisenberg uncertainty principle and the Compton wavelength formula (see equations 5 and 5b). To have zero energy requires infinite time or an infinite wavelength. Obviously the space we use for the experiment is less than infinity, and, the longest time on record is the age of the universe--also less than infinity, so yes, the energy per slit must be greater than zero.

If N equals infinity, we might naively multiply N by the average energy at each slit to get infinite energy:

But here's a better approach. First let's reset our dimension variables:

Let's express x and y in terms of the electron's initial energy:

Next, let's express x and y in terms of the average energy per slit:

To get dx we divide by the number of slits (N).

At 12 we sum all the y's. At 13 we convert that sum to its energy equivalent:

We solve the integral at 14. At 16 the infinities and epsilons cancel. That brings us to 17, the area under the curve, but expressed in terms of energy.

Multiply both sides by E^2/xy, find the square root which gives a finite energy for the electron.

Thus energy is conserved. Our final energy is the same as our initial energy instead of being absurdly infinite.