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 truely a new physical constant?
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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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If you're talking about deriving the value of Planck's constant, then no, that is not possible. The value is simply a consequence of our chosen unit system. If you're talking about deriving the fact that something analogous to Planck's constant has to exist at all, then I believe the answer is still no. To some extent that is also a consequence of our unit system, since if you use fully natural units, Planck's constant has a value of 1 and so it never shows up in the equations in the first place. But besides that, the original context in which the context was proposed was the quantization of energy, namely that the energy of an EM wave is quantized in units of $hf$. This could be considered the foundational assumption of quantum mechanics. Planck's constant is part of this assumption, so you can't really call it a derived result. |
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Maxwell assumes only that a particle has charge, not that an electron has a frequency that depends on its rest mass. So one can not deduce Planck's constant from Maxwell. de Broglie fixed that. |
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