Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold

David Elworthy, Aubrey Truman

Research output: Contribution to journalArticle

Abstract

We consider the limiting case ??0 of the Cauchy problem,?g?(x,t)/?t = 1/2?? xg?(x,t) + V(x)/?g?(x,t), with g?(x,O) = exp {-SO(x)/? } T O(x); V, SO being real-valued functions on N;TO a complex-valued function on N; V, SO, TO being independent of ?;?x being the Laplace-Beltrami operator on N, some complete Riemannian manifold. We prove some new results relating the limiting behavior of the solution to the above Cauchy problem to the solution of the corresponding classical mechanical problem D2Z(s)/?s 2 = -?ZV[Z(s)], se[O,t], with Z(t) = x and Z (O) = ?SO(Z (O)). One of our results is equivalent to the fact that for short times Schrödinger quantum mechanics on the Riemannian manifold N tends to classical Newtonian mechanics on N as ? tends to zero. © 1981 American Institute of Physics.

Original languageEnglish
Pages (from-to)2144-2166
Number of pages23
JournalJournal of Mathematical Physics
Volume22
Issue number10
Publication statusPublished - 1980

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