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.