2nd_homework

# 2nd_homework - PHY4221 Quantum Mechanic I(Fall term of 2004...

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PHY4221 Quantum Mechanic I (Fall term of 2004) Problem Set No. 2 (Answers) Question 1: (a): The normalization is quite easy: { } { } 2 2 0 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 ψ ψ π π π π φ -∞ = = Ψ = - - ÷ = + + - = + ÷ = + = 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 | 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 ψ ψ ψ ψ Ψ = = = = = Ψ (3) ( 29 ( 29 2 , ; ,0 exp 2 2 m m K x t x x x i t i t π = - - h h (4)

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( 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 π π -∞ Ψ = - - - = - - - h h h h (5) (c): because we have: ( 29 { } 2 ,0 1 exp 2 dx x a ε ε ε - Ψ = - - If a is very large, then ε could be very small for ( 29 2 ,0 1 dx x ε ε - Ψ B , this behavior is like that of Dirac delta-function, so we conclude that: ( 29 ( 29 ,0 ~ x x δ Ψ (6) So the wave function at arbitrary time t could be given by: ( 29 ( 29 ( 29 { } 2 2 , exp 2 2 exp 1 exp 2 2 x m m x t dx x x i t i t a a x m m x at i i t δ π π Ψ - - - = - B h h h h (7) And the wave function in momentum representation is also could be written down. Thus in this
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