An Amplitude Squared Equals a Probability--a Mathematical Proof

Start with a wave:

Add more waves. Waves are in and out of phase with each other (constructive and destructive interference):

The range is from zero degrees out of phase to 180 degrees out of phase. On average, any pair of waves is 90 degrees out of phase, making a sine wave and a cosine wave:

Each has an amplitude (Ao). We can add the waves together, making a complex number (equation 1). Since our intention is to isolate and square the amplitude, we need a complex conjugate (equation 2). Using Euler's identity we write the following equations:

When we square the the left sides and square the right sides of equations 1 and 2, here's what we get:

It's entirely possible we don't get a probability. Instead, we could get a value greater than one. Plus we have distance units--so we need to normalize the amplitude:

Now, let's express our oscillating wave(s) in terms of Hooke's law and derive an energy (E):

Below we make a couple of substitutions to derive equation 10:

At 11 we define the probability of E. It could be from zero to one:

Multiply both sides of equation 10 by the probability P(E) then add a pinch of algebra to get equation 16:

Equation 16 shows the probability is equal to the square of the reduced normalized amplitude. Of course we are not limited to just energy eigenvalues. We can show that any type of eigenvalue has a probability equal to the square of a normalized amplitude:

In fact, if we express energy E in terms of momentum (p), mass (m), position (x), time (t) and wave number (k), we discover that the momentum, mass, position, time and wave number each have the same probability as the energy: the square of the amplitude A'. This is due to the energy being dependent on a specific value of each of these other eigenvalues. Each term below contains one eigenvalue and constants (which don't change). So the energy eigenvalue correlates with each of these other eigenvalues, and, likewise, the probabilities correlate.

In conclusion it is safe to say a wave amplitude squared does not guarantee a probability; however, it looks as though the square of a normalized amplitude does.