Abstract
It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrödinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) “practical calculations” — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrödinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in “practical” calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.
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References
R. P. Feynman,Rev. Mod. Phys. 20, 367 (1948).
R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
E. Nelson,J. Math. Phys. 5, 332 (1964).
R. E. L. Monaco, Ph. D. Dissertation, Inst. Física “Gleb Wataghin” UNICAMP, 1992.
L. S. Schulman,Techniques and Applications of Path Integration (Wiley, New York, 1981).
F. E. Wiegel,Introduction to Path-Integrals Methods in Physics and Polimer Sciences (World Scientific, Singapore, 1986).
J. W. Negele and H. Orland,Quantum Many-Particle Systems (Addison-Wesley, Reading, 1988).
S. Lundqvist, A. Ranfagni, V. Sa-yakanit and L. S. Schulman, eds.,Path Integral Method with Applications (Word Scientific, Singapore, 1987).
H. H. Torriani,Adv. in Appl. Math. 8, 345 (1987).
A. K. Kapoor,Phys. Rev. D 30, 1750 (1984).
R. L. Monaco and W. A. Rodrigues, Jr.,Phys Lett. A 179, 235 (1993).
R. L. Monaco and E. Capelas de Oliveira,J. Math. Phys. 35, 637 (1994).
R. P. Feynman, R. B. Leighton, and M. Sands,Quantum Mechanics: The Feymann Lectures on Physics, Vol. 3 (Addison-Wesley, Reading, 1965).
Franco Selleri,Le grand débat de la Théorie quantique (Flammarion France, 1986).
L. de Brolie,C.R. Acad. Sc. (Paris) 183, 447 (1926).
D. Bohm,Phys. Rev. 85, 166 (1952).
R. L. Monaco and G. G. Cabrera,Found. Phys. Lett. 7, 1 (1994).
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Monaco, R.L., Lagos, R.E. & Rodrigues, W.A. A riemann integral approach to Feynman's path integral. Found Phys Lett 8, 365–373 (1995). https://doi.org/10.1007/BF02187816
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DOI: https://doi.org/10.1007/BF02187816