Abstract
In this paper we unravel the connection between the quantum mechanical formalism and the Central limit theorem (CLT). We proceed to connect the results coming from this theorem with the derivations of the Schrödinger equation from the Liouville equation, presented by ourselves in other papers. In those papers we had used the concept of an infinitesimal parameter δx that raised some controversy. The status of this infinitesimal parameter is then elucidated in the framework of the CLT. Finally, we use the formal apparatus developed in our previous papers and the results of the present one to advance an alternative objective interpretation of quantum mechanics in which its relations with the classical framework are made explicit. The relations between our approach and those using the Wigner–Moyal transformation are also addressed.
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L. S. F. Olavo, Physica A 262, 197(1999).
L. S. F. Olavo, Physica A 271, 260(1999).
L. S. F. Olavo, Phys. Rev. A 61, 052109(2000).
R. P. Feynmanand A. R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York 1965).
D. Kershaw, Phys. Rev. B 136, 1850(1964), G. G. Comisar, Phys. Rev. B 138, B1332 (1965). T. W. Marshall, Proc. Cambridge Philos. Soc. 61, 537 (1965). E. Nelson, Phys. Rev. 150, 1079 (1966), L. de la Peña, J. Math. Phys. 10, 1620 (1969), L. de la peña, Phys. Lett. A 31, 403 (1970). L. de la Peña and A. M. Cetto, Phys. Rev. D 3, 795 (1971), to cite but a few.
E. Wigner, Phys. Rev. 40, 749(1932).
J. E. Moyal, Cambridge Phil. Soc. 45, 99(1949).
T. Takabayasi, Prog. Theor. Phys. 11, 341(1954).
P. A. M. Dirac, Rev. Mod. Phys. 17, 195(1945). T. Takabayasi, Prog. Theor. Phys. 11, 341 (1954). J. C. T. Pool, J. Math. Phys. 7, 66 (1966). L. Cohen, J. Math. Phys. 7, 781 (1966). S. P. Misra and T. S. Shankara, J. Math. Phys. 9, 299 (1968). G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161 (1970). N. D. Cartwright, Physica A 83, 210 (1976). R. F. O'Connell and E. P. Wigner, Phys. Lett. A 85, 121 (1981), and many others.
Cf., for example, with expressions (3.60a,b,c), p. 163, inR. L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions (Prentice Hall, New Jersey, 1990), pp.152-163.
A. I. Kinchin, Mathematical Foundations of Statistical Mechanics, (Dover, New York 1949) pp. 8485, translated from the Russian by G. Gamow (we changed slightly Khinchin's notation to fit our own).
P. L. Meyer, Introductory Probability and Statistical Applications (Addison-Wesley, New York,1965), pp. 270ff(where the restricted case we are considering is better explained).
P. Levy, Théorie des erreurs. La loi de Gauss et les lois exceptionelles. in Oeuvres de Paul Levy, (Ecole Polytechnique, France 1976), pp. 14-49 P. Levy shows in his own derivation that it is not necessary to make the hypothesis that the characteristic function may be decomposable into a Taylor series; indeed, his derivation is much more general than the one presented by us here. However, the present derivation has the advantage of being known to the great majority of physicists, since it is part of usual statistical mechanics' manuals.
In paper [2] we have imposed the restriction (∂S/∂x) Δx = 0 = 0, but this restriction is not necessary.
L. E. Ballentine, Rev. Mod. Phys. 42, 358(1970). W. Heisenberg, "The development of the interpretation of the quantum theory," in Niels Bohr and the Development of Physics. W. Pauli, ed. (McGraw-Hill, New York, 1955), p. 12. T. Brody, The Philosophy behind Physics (Springer, Berlin, 1993). A. Landé, From Dualism to Unity in Quantum Mechanics (Cambridge University Press, Cambridge, 1960), and many others (cf. the references in Ballentine's paper, for instance).
F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, Singapore 1965).
M. Born, Natural Philosophy of Cause and Chance (Oxford University Press, London 1949), p. 94.
W. L. Scott, The Conflict between Atomism and Conservation Theory: 1644 to 1860 (MacDonald, London, andElsevier, New York, 1970).
J. Mehra, Found. Phys. 17, 461(1987).
J. R. Shewell, Am. J. Phys. 27, 16(1959).
D. Alonso, J. G. Muga,and R. SalaMayato, Phys. Rev. A 64, 016101(2001).
R. P. Feynman, Statistical Mechanics, a Set of Lectures, (Advanced Book Classics), (Addison-Wesley, Massachussets, 1998), p. 72ff.
L. Cohenand Y. I. Zaparovanny, J. Math. Phys. 21, 794(1980).
N.L. Balazs, Physica A 102, 236(1980).
C. Lanczos, The Variational Principles of Mechanics, (Dover, New York 1986), p. 201ff.
H. W. Lee, Phys. Rep. 259, 147(1995).
M. Hillery, R. F. O'Connell, M. O. Scullyand E. P. Wigner, Phys. Rep. 106, 121(1984).
R. G. Parrand W. Yang, Density-Functional Theory of Atoms and Molecules, (Oxford University Press, New York 1989).
N. R. Hanson, Am. J. Phys. 27, 1(1959). A. Shimony, Am. J. Phys. 31, 755 (1963). E. E. Witmer, Am. J. Phys. 35, 40 (1963). E. P. Wigner, Am. J. Phys. 31, 6 (1963). P. Pearle, Am. J. Phys. 35, 742 (1967). J. P. Wesley, Found. Phys. 14, 155 (1984), and many others.
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Olavo, L.S.F. Foundations of Quantum Mechanics: The Connection Between QM and the Central Limit Theorem. Foundations of Physics 34, 891–935 (2004). https://doi.org/10.1023/B:FOOP.0000034222.65544.26
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DOI: https://doi.org/10.1023/B:FOOP.0000034222.65544.26