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I was watching a video about fundamental Quantum Mechanics. The video lecture is excellent but I am confused after hearing the part at the 9.46 sec of this video . He said repeatedly from the previous video that the probability amplitude can not be proved but the MAX BORN asserted it from experiment. My question is why the probability aptitude AA* is assertion from an experiment but not an theoretical argument.

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Generally speaking nothing in physics can be "proved" in the sense of mathematics. What can be done is to compare measurements with predictions of a theory. QM basically tells us that if we take a magical quantity $\psi$ and evolve that with a magical Operator $H$ in the magical Schroedinger equation and then apply another magical operator $x$ or $p$ to that, then we get a number which can be compared with the AVERAGE reading on a dial labeled "x" or "p" of a specific quantum mechanical experiment that has been repeated many times over.

And if you do that right in the lab, you will be jumping of joy, because the magical numbers and the measured ones seem to agree with an astonishing precision.

Please note that this does not fundamentally differ from what we do in classical mechanics, except that we lack a sense of wonder about why mass and force and acceleration are related trough Newton's second law. The reason why we lack that is because it is absolutely tantamount for a kid to learn that relationship biologically while it learns to walk. Kids who biologically still keep wondering about $F=ma$ at the time they are 15 won't make it into the high school football team and they will have a significantly smaller chance to marry the head cheerleader... hence the reason why Newtonian mechanics feels familiar and QM does not.

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ψ ψ* is the probability density not the Amplitude which is ψ, this is the standard belief of most "quantum mechanics" because a self consistent theory can be constructed from this. It is not derived from standard assumptions like the probability of rolling dice.
Some famous Physicists hated this idea, most notably Schrodinger who invented the "Schrodinger's Cat" example to try to show the absurdity of the notion. Schrodinger believed ψ ψ* was proportional to the electron charge density. Another skeptic was Einstein, who famously said that God didn't play dice.

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