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Unformatted text preview: PHY4221 Quantum Mechanic I (Fall term of 2004) Problem Set No. 2 (Answers) Question 1: (a): The normalization is quite easy: { } { } 2 2 2 1 exp 2 2 exp 2 dxA a x dxA ax A a  = = = (1) Thus we have A a = . Then we have: ( 29 ( 29 ( 29 2 3 2     exp exp 2 1 1 2 2 2 2 1 ,0 p dx p x x dx p x x a px dx i a x a p p a i a i a a p a a a k k  = = = = + + = + = + = h h h h h h h h (2) (b): Using the propagator, it is easy to obtain: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ,   ,0  ,0   , ; ,0  , ; ,0 ,0 x t x t x U t dx x U t x x dxK x t x x dxK x t x x = = = = = (3) ( 29 ( 29 2 , ; ,0 exp 2 2 m m K x t x x x i t i t = h h (4) ( 29 ( 29 { } ( 29 2 2 , exp exp 2 2 exp 2 2 m m x t dx x x a a x i t i t ma m dx x x a x i t i t  = = h h h h (5) (c): because we have: ( 29 { } 2 ,0 1 exp 2 dx x a  =  If a is very large, then could be very small for ( 29 2 ,0 1 dx x  B , this behavior is like that of Dirac deltafunction, so we conclude that: ( 29 ( 29 ,0 ~ x x (6) So the wave function at arbitrary time t could be given by: ( 29...
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