ps09 - Homework (9) for Physics 316, Modern Physics I

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Unformatted text preview: Homework (9) for Physics 316, Modern Physics I (Hoffstaetter/Drasco/Thibault) Due Date: Friday, 04/08/05 - 9:55 in 132 Rockefeller Hall Exercise 1: In Electrodynamics the charge density ( ~x, t ) and the current density ~ j ( ~x, t ) satisfy a con- tinuity equation, t ( ~x, t ) + ~ ~ j ( ~x, t ) = 0 . (1) Show that the probability density = | ( ~x, t ) | 2 satisfies the same continuity equation together with the probability current density ~ j = h m Im { ( ~x, t ) * ~ ( ~x, t ) } . Exercise 2: Given a wave function ( x, t ) at t = 0 as (a) ( x, 0) = A a e- x 2 2 and (b) ( x, 0) = A b e- | x | , where A a and A b are chosen in each case to normalize the wave function to 1. For each case, find the momentum representation of the wave function, i.e. find F ( k ) so that ( x, t ) can be written as ( x, t ) = Z - F ( k ) e i [ kx- ( k ) t ] 2 dk . (2) Exercise 3: Given a wave function ( x, t ) at t = 0 as ( x, 0) = A a e...
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