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Unformatted text preview: Physics 731 Assignment #7, Solutions 1. The simplest way to compute the propagator is to exploit the fact that for the linear potential, K ( x,t ; x , 0) = A ( t ) e iS cl / ¯ h , in which S cl is the classical action and A ( t ) = m 2 πi ¯ ht 1 / 2 , as for the free particle. S cl can be computed by solving the classical equations of motion: x ( t ) = x + p t m + Ft 2 2 m , where p is the initial momentum and x is the initial position, and using this trajectory in the action S cl = Z t Ldt = Z t dt 1 2 m ˙ x 2 + Fx . As ˙ x ( t ) = p /m + Ft/m , we have S cl = Z t dt 1 2 m ( p 2 + 2 p Ft + F 2 t 2 ) + Fx + Fp t m + F 2 t 2 2 m = p 2 t 2 m + p Ft 2 m + Fx t + F 2 t 3 3 m . Since p = m ( x x ) /t Ft/ 2 , the classical action takes the form S cl = m ( x x ) 2 2 t + Ft ( x + x ) 2 F 2 t 3 24 m , and hence the propagator is K ( x,t ; x , 0) = m 2 πi ¯ ht 1 / 2 exp i ¯ h m ( x x ) 2 2 t + Ft ( x + x ) 2 F 2 t 3 24 m ....
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This note was uploaded on 12/14/2009 for the course QUANTUM I 731 taught by Professor Everett during the Fall '09 term at University of Wisconsin.
 Fall '09
 Everett

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