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Heck, in Newton's case, you'd have to explain electrostatics before you could even get started on QM, and electrodynamics and electromagnetism before you could get very far. And I would claim that electromagnetism in particular is impossible to understand until and unless you're familiar with PDEs, because you have to be able to quantify the positive and negative feedbacks in order to even give a qualitative description of the system's behaviour.
Friends come and go. Enemies accumulate.
I would disagree. There is an entire conceptual apparatus that developed with
17th-19th century mathematical physics - generator functions, matrix algebra,
vector calculus - etc. I don't see how you can make meaningful predictions in
QM outside that framework.
Electrostatics is actually a bad example to use, precisely because the theory is mathematically identical to
Newtonian gravity. Even relativity would have been no problem to these guys. Einstein's contribution, while crucial,
is technically really very small, as it amounts to doing hyperbolic geometry instead of the Euclidean one. Newton
knew more about conics than most mathematicians probably do today.
As to making useful predictions in QM without these methods, remember that matrix mechanics is only Heisenberg's
picture. The Schroedinger picture is about wave equations, which had already been worked out in the middle of the
17th century by Euler and the Bernoulli gang.
$E(X_t|F_s) = X_s,\quad t > s$
Netwon would have been a pig to persuade, but I doubt he would have had problems with the theory. Every year hundreds of ordinarily talented undergrads work their way through the basics without falling off anything tall and hurting themselves, so a genius really wouldn't find it difficult.
Ed Witten of string theory fame apparently worked through an entire three year undergrad physics curriculum over a summer holiday - competently enough to enrol as an applied maths postgrad, even though his original major was history, and he was planning to be a political journalist.
He may have had help from his father, who was another theorist. But even so.
He also lasted one term as an economics wannabe, which may or may not say something relevant and interesting about economics.
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