[talanum1 (OP)]

We have real waves in the double slit experiment, but in order to be consistent we must make them identical to their probability waves. What is the wave amplitude squared?

[Bored chemist]

Did you think that made sense?

[Kartazion]

What is the wave amplitude squared?

A wave function doesn't in particular represent amplitude of a wave. The square of the wave function gives the probability to find the particle with the energy at a certain position in space.

To check.

[thebrain13]

Read the book QED the strange theory of light and matter. Richard Feynman outlines how essentially how all probability works from the Heisenberg uncertainty principle to why light bounces off a mirror in a straight line to lenses. He wins the Nobel prize largely because of this. If you want the cliff notes version, google image search Feynman QED and look for a picture of a lens and a bunch of arrows and the caption figure 36 so you can see a visual of what I'm talking about. This is a good exercise for anybody who wants a visual understanding of quantum probability.

What he does is he sums up every possible combination of where a particle could go by adding a bunch of arrows together. These arrows are represented like a clock hand. Corresponding to the peak of a photon the clock hand is at 12 and at its trough at 6. Add up all these arrows and square them and you get the probability of where the photon will go. For instance, if there was only two paths to the same spot, one at 12 and one at 6 the probability of it going there would be zero because the arrows would cancel each other out.

If there were two paths to one spot each corresponding at 12 then he would add the two arrows/vectors together  and square them to get the probability. This is what squaring the amplitude means. Of course, to do this completely we must some up equally every possible outcome together. This makes it messy. I was never taught this in undergrad for instance, because it would be a messy exam question but adding and squaring these amplitudes in different scenarios is theoretically how ALL quantum probabilities are calculated.

In regards to whether the waves are "real" or not. Well there is some ambiguity in what you mean. Feynman would be diplomatic and say he doesn't know why this works, it just does. Of course, Feynman knew many different camps of thought. Like American Universities, quantum theorists and Einstein. Einstein actually stood up for Feynman's theory, probably why it stands today. He writes about this in one of his books, Surely you're joking Mr. Feynman, or what do you care what other people think.

Although I can not confirm it, just based on my senses of Einstein. Einstein would of liked the concept of a mathematics of quantum mechanics being represented by "spread out" visuals like the infinite paths of the path integral of QED because it would be easy for him to visualize how to use "field" interpretations of particulate matter which I know he was a big fan of.

Einstein took a liking to Louis de Broglie as he was focused on the wave like nature of matter. Pre Einstein there was no concept of matter being waves. Louis de Broglie liked the concept of waves "piloting" matter hence the connections.

Here is a quote from Einstein's book, the evolution of physics page 276. "It was previously shown that there are phenomena in which light reveals its wave-like character and others in which light reveals its corpuscular character. After becoming used to the idea that light is a wave, we found, to our astonishment, that in some cases, for instance in the photoelectric effect, it behaves like a shower of photons. Now we have just the opposite state of affairs for electrons. We accustomed ourselves to the idea that electrons are particles, elementary quanta of electricity and matter. Their charge and mass were investigated. If there is any truth in de Broglie's idea, then there must be some phenomena in which matter reveals its wave-like character."

Einstein had a vision that all physics could be fields, and this is one of many examples. If you want to view the wavefunction as something "real" you could view the twin slits in terms of fields traveling through both slits, with some sort of entangled force bringing them back together. Not many people who are still alive will take you seriously but if you ask me it certainly could be. Perhaps a physical pilot field as was suggested by de Broglies is driving the field like qualities of Q.M. and particles. Perhaps the helix shaped circularly polarized E.M. field drives the physical shape of DNA and the many helices in how it is coiled. Nobody will take you seriously, but it could be. To most people though, those probabilistic wavefunctions are just wiggly lines on a piece of paper. Does that answer your question?