Quiz1 - PHY4221 Quantum Mechanics I Fall Term of 2005 Quiz...

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PHY4221 Quantum Mechanics I Fall Term of 2005 Quiz 1 (Suggested Solution) The eigen functions of the infinite square-well potential are () 2 sin 0 0o t h e r w i s e n nx x a x aa π φ ⎡⎤ << ⎢⎥ = ⎣⎦ , (1) where we have assumed that the well lies in 0<x<a; the corresponding eigen-energies are: 222 2 ,1 , 2 , 3 , 2 n n En ma == = (2) According the expression of propagator ()( ) ( ) ( ) ,; , exp n nn n Ett Kx txt x x i φφ ′′ =− = , we have: 22 2 1 2 sin sin exp 0 , 2 t h e r w i s e n i n tt x x a aaa ma ππ = −< < ′′ = = . (3) At time t`=0, the wave function of a particle in this well is given as: () () 12 ,0 3 x xx Ψ= + (4) which, after normalization, is: 31 x + . (5) Then at an arbitrary later time, the wave function is given by: () ( )( ) () ( ) 13 2 1 11 1 33 3 2 2 ,, ; , 0 , 0 exp 2 9 exp exp exp 2 n xt dxK xtx x in dx x x x x t ma ii dx x x x t x x x t ma ma
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