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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.
There's nothing specifically QM about any of your examples (generating functions, matrices or vector calculus). Historically, these ideas were developed for entirely different areas of mathematics in the 19th century or earlier, but even that late arrival had no limiting effect at all on scientists' ability to solve complex real world problems much earlier. For example, Euler (as for that matter Newton) was perfectly capable of treating full 3d motion in the middle of the 18th century without requiring the crutch of matrices or vectors. Monge was a master of PDE theory - in 1795!

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.

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$E(X_t|F_s) = X_s,\quad t > s$

by martingale on Mon Jun 1st, 2009 at 05:47:47 AM EST
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