Can mathematics (including statistics, dynamical systems,...) combined with classical electromagnetism (using only the constants appearing in chargefree Maxwell equations) be used to derive the Planck constant? Can it be proven that Planck's constant is truly a new physical constant?
Answer
Look, Dr. Zaslavsky is completely correct. But. The great mathematician Jean Leray once, after being asked to think about Maslov's work on asymptotic methods to approximate the solutions of partial differential equations which were generalisations of the WKB method, decided, in the 70's, to write an entire book titled Lagrangian Analysis and Quantum Mechanics, note he gives his own special meaning to « Lagrangian Analysis.», MIT Press, see the nice abstract entitled « The meaning of Maslov's asymptotic method: The need of Planck's constant in mathematics.»
This is not a derivation of the magnitude of Planck's constatnt from Maxwell's equations, but it is a profound motivation for why there should be some finite, small, constant such as Planck's from the standpoint that the caustics you get in geometrical optics cannot be physical, and yet geometric optics ought to be a useful approximation to wave optics. From this point of view, there ought to be some constant like Planck's constant, at least in pure mathematics.
It is, however, very advanced: inaccessible unless you already know about Fourier integral operators in Symplectic manifolds, such as in Duistermaat's book or Guillemin and Sternberg, Symplectic Techniques in Physics. Maslov's original book is, although non-rigorous, very insightful and more accessible.
For a physicist, though, perhaps just the basics of the Hamiltonian relationship between geometrical optics and wave optics, and the basics of the WKB method, would be more important.
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