PHY4221 Quantum Mechanics I Fall Term of 2005 Quiz 1 (Suggested Solution) The eigen functions of the infinite square-well potential are ()2sin00otherwisennxxaxaaπφ⎧⎡⎤<<⎪⎢⎥=⎨⎣⎦⎪⎩, (1) where we have assumed that the well lies in 0<x<a; the corresponding eigen-energies are: 2222,1,2,3,2nnEnma==⋅⋅⋅=(2) According the expression of propagator()()()( ),; ,expnnnnEttKxtxtxxiφφ∗′−⎡⎤′′′=−⎢⎥⎣⎦∑=, we have: 22212sinsinexp0,2therwiseninttxx aaaamaππ∞=⎧′⎡⎤′′−−<<⎪⎢⎥′′ =⎣⎦⎨⎪⎩∑=.(3)At time t`=0, the wave function of a particle in this well is given as: () ()12,03xxxΨ=+(4) which, after normalization, is: 31x+. (5) Then at an arbitrary later time, the wave function is given by: () ( )( )() ( )132111133322,,;,0,0exp29expexpexp2nxtdxK xtxxindxxxxxtmaiidxxxxtxxxtmama
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