Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation
- 1 December 1977
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 18 (12) , 2308-2315
- https://doi.org/10.1063/1.523240
Abstract
We consider the limiting case λ→0 of the Cauchy problem ∂uλ/∂t= (λ/2μ) ∇2xuλ +[V (x)/λ]uλ, uλ(x,0) =exp[−S0(x)/λ]T0(x); S0, T0 independent of λ, for both real and pure imaginary λ. We prove two new theorems relating the limiting solution of the above Cauchy problem to the corresponding equations of classical mechanics μ (d2x/dτ2)(τ) =−∇xV[x (τ)], τ∈ (0,t). These relationships include the physical result quantum mechanics → classical mechanics as h/→0.This publication has 14 references indexed in Scilit:
- The semiclassical expansionAnnals of Physics, 1976
- DIFFUSION PROCESSES AND RIEMANNIAN GEOMETRYRussian Mathematical Surveys, 1975
- Feynman path integralsCommunications in Mathematical Physics, 1974
- Feynman's path integralCommunications in Mathematical Physics, 1972
- Derivation of the Schrödinger Equation from Newtonian MechanicsPhysical Review B, 1966
- THE WIENER INTEGRALRussian Mathematical Surveys, 1963
- On distributions of certain Wiener functionalsTransactions of the American Mathematical Society, 1949
- Transformations of Wiener integrals under a general class of linear transformationsTransactions of the American Mathematical Society, 1945
- The Correspondence Principle in the Statistical Interpretation of Quantum MechanicsProceedings of the National Academy of Sciences, 1928
- Eine Verallgemeinerung der Quantenbedingungen f r die Zwecke der WellenmechanikThe European Physical Journal A, 1926