The Hamilton-Jacobi equation of classical mechanics1 is generally obtained in the limit 0 from the time-dependent Schrödinger equation as follows.2,3 If ψ(r,t) is the complex wave function of a single particle of mass m in a potential V(r), the solution of the time-dependent Schrödinger equation,

iψ(r,t)t=22m2ψ(r,t)+V(r)ψ(r,t),
(1)

is written in the form

ψ(r,t)=R(r,t)exp[iS(r,t)],
(2)

where R(r,t) and S(r,t) are real functions of the position r and...

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