Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)We apply the influence-functional method of Feynman and Vernon to the study of Brownian motion at arbitrary temperature. By choosing a specific model for the dissipative interaction of the system of interest with its environment, we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient. We show that in the limit h→0 the results obtained from the influence functional formalism reduce to the classical Fokker-Planck equation. In the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature, we obtain an explicit expression for the time evolution of the complete density matrix ρ{variant}(x, x′, t) when the system starts in a particular kind of pure state. We compare our results with those of other approaches to the problem of dissipation in quantum mechanics. © 1983.12135876162013/15478-3; FAPESP; São Paulo Research Foundation; 2013/23396-7; FAPESP; São Paulo Research FoundationFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Wax, (1954) Selected Papers on Noise and Stochastic Processes, , Dover, New YorkKanai, On the Quantization of the Dissipative Systems (1948) Progress of Theoretical Physics, 3, p. 440Louisell, (1973) Quantum Statistical Properties of Radiation, , Wiley, New YorkKostin, (1972) J. Chem. Phys., 57, p. 3589Nelson, (1966) Phys. Rev., 150, p. 1079Yasue, (1978) Annals of Physics, 114, p. 479Dekker, Quantization of the linearly damped harmonic oscillator (1977) Physical Review A, 16, p. 2116Senitzky, (1960) Phys. Rev., 119, p. 670Mori, Transport, Collective Motion, and Brownian Motion (1965) Progress of Theoretical Physics, 33, p. 423Zwanzig, (1960) J. Chem. Phys., 33, p. 1338Nakajima, On Quantum Theory of Transport Phenomena (1958) Progress of Theoretical Physics, 20, p. 948Haken, (1975) Rev. Mod. Phys., 47, p. 67Haake, Quantum Statistics in Optics and Solid State Physics (1973) Springer Tracts in Modern Physics, 66. , Springer, BerlinKoch, Van Harlingen, Clarke, (1980) Phys. Rev. Lett., 45, p. 2132Benguria, Kac, (1981) Phys. Rev. Lett., 46, p. 1Ford, Kac, Mazur, (1965) J. Math. Phys., 6, p. 504Iche, Nozieres, (1978) Physica, 91 A, p. 485Davies, (1976) Quantum Theory of Open Systems, , Academic Press, New YorkSchwinger, (1961) J. Math. Phys., 2, p. 407Zwanzig, (1973) J. Stat. Phys., 9, p. 215A.O. Caldeira and A.J. Leggett, to appear in Annals of PhysicsFeynman, Hibbs, (1965) Quantum Mechanics and Path Integrals, , McGraw-Hill, New YorkFeynman, Vernon, (1963) Annals of Physics, 24, p. 118Forster, (1975) Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, , Benjamin, New YorkMorse, Feshbach, (1953) Methods of Theoretical Physics, , McGraw-Hill, New York, sect. 3.2Kjeldysh, (1965) JEPT-Sov. Phys., 20, p. 1018Nemes, de Toledo Piza, (1977) Revista Brasileira de Física, 7, p. 261Papadopoulos, Functional integrals in Brownian motion (1968) Journal of Physics A: General Physics, 1, p. 413Wigner, (1932) Phys. Rev., 40, p. 749Berry, Semi-Classical Mechanics in Phase Space: A Study of Wigner's Function (1977) Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 287, p. 237Caldeira, (1980) Ph.D. Thesis, , University of Sussex, unpublishedLandau, Lifshitz, (1969) Statistical Physics, p. 393. , Pergamon, LondonTikochinsky, Exact propagators for quadratic Hamiltonians (1978) Journal of Mathematical Physics, 19, p. 888Feshbach, Tikochinsky, A Festschrift for I.I. Rabi (1977) Trans. New York Ac. Sc. Ser. 2, 38, p. 44Dekker, (1981) Phys. Rep., 80, p. 1Dekker, (1979) Physica, 95 A, p. 311Agarwal, (1971) Phys. Rev. A, 4, p. 739Svin'in, (1976) Theor. Math. Fiz., 27, p. 270