Consider the nonrelativistic Schrodinger equation for a particle moving in a time dependent, 3-dimensional potential well: (a)
Question:
Consider the nonrelativistic Schrödinger equation for a particle moving in a time dependent, 3-dimensional potential well:
(a) Seek a geometric-optics solution to this equation with the form ψ = AeiS/ℏ, where A and V are assumed to vary on a length scale L and timescale T long compared to those, 1/k and 1/ω, on which S varies. Show that the leading order terms in the two-length scale expansion of the Schrödinger equation give the Hamilton-Jacobi equation
Our notation φ ≡ S/ℏ for the phase φ of the wave function ψ is motivated by the fact that the geometric-optics limit of quantum mechanics is classical mechanics, and the function S = ℏφ becomes, in that limit, “Hamilton’s principal function,” which obeys the Hamilton-Jacobi equation.(b) From Eq. (7.52a) derive the equation of motion for the rays (which of course is identical to the equation of motion for a wave packet and therefore is also the equation of motion for a classical particle):
where p = ∇S.
(c) Derive the propagation equation for the wave amplitude A and show that it implies
Interpret this equation quantum mechanically.
Step by Step Answer:
Modern Classical Physics Optics Fluids Plasmas Elasticity Relativity And Statistical Physics
ISBN: 9780691159027
1st Edition
Authors: Kip S. Thorne, Roger D. Blandford