# From Classical Hamiltonian Flow to Quantum Theory: Derivation of the Schrödinger Equation

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## Abstract

It is shown how the essentials of quantum theory, i.e., the Schrödinger equation and the Heisenberg uncertainty relations, can be derived from classical physics. Next to the empirically grounded quantisation of energy and momentum, the only input is given by the assumption of fluctuations in energy and momentum to be added to the classical motion. Extending into the relativistic regime for spinless particles, this procedure leads also to a derivation of the Klein-Gordon equation. Comparing classical Hamiltonian flow with quantum theory, then, the essential difference is given by a vanishing divergence of the velocity of the probability current in the former, whereas the latter results from a much less stringent requirement, i.e., that only the average over fluctuations and positions on the average divergence be identical to zero.

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## REFERENCES

- 1.H. Goldstein,
*Classical Mechanics*(Addison-Wesley, Reading, MA, 1950).zbMATHGoogle Scholar - 2.P. R. Holland, “Electromagnetism, particles and anholonomy,”
*Phys. Lett. A***91**(1982), 275–278.ADSMathSciNetCrossRefGoogle Scholar - 3.E. Schr\:odinger, “Unsere Vorstellung von der Materie” (1952), in E. Schr\:odinger,
*Was ist Materie?*, Audio CD-ROM, K\:oln, 2002 (our translation).Google Scholar - 4.S. A. Werner, R. Clothier, H. Kaiser, H. Rauch, and H. W\:olwitsch, “Spectral filtering in neutron interferometry,”
*Phys. Rev. Lett.***67**(1991), 683–687.ADSCrossRefGoogle Scholar - 5.D. L. Jacobson, S. A. Werner, and H. Rauch, “Spectral modulation and squeezing at high-order neutron interferences,”
*Phys. Rev. A.***49**(1994), 3196–3200.ADSCrossRefGoogle Scholar - 6.M. J. W. Hall and M. Reginatto, “Schr\:odinger equation from an exact uncertainty principle,”
*J. Phys. A***35**(2002), 3289–3303.ADSMathSciNetCrossRefzbMATHGoogle Scholar - 7.R. J. Harvey, “Navier-Stokes analog of quantum mechanics,”
*Phys. Rev.***152**(1966), 1115.ADSCrossRefGoogle Scholar - 8.D. Bohm, “A suggested interpretation of the quantum theory in terms of \lshidden variables\rs,” I and II,
*Phys. Rev.***85**(1952), 166–179, 180–193.ADSMathSciNetCrossRefGoogle Scholar - 9.P. R. Holland,
*The Quantum Theory of Motion*(University Press, Cambridge, 1993).CrossRefGoogle Scholar - 10.R. A. Fisher, “Theory of statistical information,”
*Phil. Trans. Roy. Soc. London, Ser. A***222**(1930), 309–368.ADSCrossRefGoogle Scholar - 11.T. M. Cover and J. A. Thomas,
*Elements of Information Theory*(Wiley, New York, 1991).CrossRefzbMATHGoogle Scholar - 12.B.-G. Englert, M. O. Scully, and H. Walther, “Complementarity and uncertainty,”
*Nature***375**(1995), 367.ADSCrossRefGoogle Scholar - 13.R. P. Feynman, R. B. Leighton, and M. Sands,
*The Feynman Lectures on Physics*, Vol. II (Addison-Wesley, Reading, MA, 1965), Chap. 40–2.zbMATHGoogle Scholar - 14.E. Nelson,
*Quantum Fluctuations*(University Press, Princeton, 1985).zbMATHGoogle Scholar - 15.P. Garbaczewski, “Noise perturbations in the Brownian motion and quantum dynamics,”
*Phys. Lett. A***257**(1999), 31–36.ADSCrossRefGoogle Scholar - 16.G. Gr\:ossing,
*Quantum Cybernetics. Toward a Unification of Relativity and Quantum Theory via Circularly Causal Modeling*(Springer, New York, 2000).Google Scholar - 17.G. Gr\:ossing, “Quantum cybernetics: A new perspective for Nelson's stochastic theory, nonlocality, and the Klein-Gordon equation,”
*Phys. Lett. A***196**,1 (2002), 1–8.ADSMathSciNetCrossRefGoogle Scholar - 18.R. S. Mulliken, “The band spectrum of boron monoxide,”
*Nature***114**(1924), 349–350. This paper is usually considered as reporting the first empirical evidence of the reality of the zero-point energy; many others were to follow. Note that the paper appeared well before the advent of the complete formulation of modern quantum theory.ADSCrossRefGoogle Scholar