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Unformatted text preview: Problem Set 09 Note: This problem set is due Nov 15 before midnight. Please leave it in my mailbox located at the first floor of Rutherford building. 1. Recall the Schr¨ odinger equation that we derived in the class. Its an equation with first order time derivative and second order spatial derivatives. There also exist another equation called the continuity equation in hydrodynamics which is given by ∂ρ ∂t + ∇ · J = , (1) where ρ is the charge density, and J is the current. The equation (1) implies that the rate of flow of any kind of charge is equivalent to some current, so that the total flux is conserved over any cross-sectional area. Show that the Schr¨ odinger equation is exactly the charge conservation equation (1) if we identify the charge ρ with the probability density | ψ | 2 in quantum mechanics and the current J with the following quantity 1 : J ≡ - i 2 m ψ * ∂ψ ∂x- ψ ∂ψ * ∂x . (2) For a free particle with momentum p and energy E , can you determine the corresponding...
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This note was uploaded on 02/20/2011 for the course PHYS 357 taught by Professor Keshavdasgupta during the Fall '05 term at McGill.
- Fall '05