Physics 741-742

Physics 741-742 - Physics 741 Graduate Quantum Mechanics 1...

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Physics 741 – Graduate Quantum Mechanics 1 Solution Set J Due Friday, September 26 1. [15] A particle of mass m lies in a one-dimensional infinite square well in the region [0, a ]. At t = 0, the wave function in the allowed region is given by () ( ) 52 03 0 x t a ax x Ψ= = (a) [5] Write the wave function in the form ( ) 0 nn n tc ψ where n are the energy eigenstates. Write the wave function t Ψ at all times. The integral we need is () ( ) ( ) *5 2 00 0, 0 2 3 0 s i n aa n ct x x t d x a a a x x n x a d x ψψ π = = Ψ= = ∫∫ This isn’t too hard to do ourselves, but let’s let Maple do the work. > assume(n::integer);integrate((a*x-x^2)*sin(Pi*n*x/a), x=0. .a)/a**3; 33 26 0 960 if odd 11 0i f e v e n n n c n n ⎡⎤ =− = ⎣⎦ At arbitrary time, then the wave function is ( ) ( ) 2 2 2 odd exp 960 exp 2 n n i E t n i n t m a ψπ = ∑∑ == (b) [5] Check that the wave function is properly normalized, both in the original coordinate basis, and in the new basis, either analytically or numerically.
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Physics 741-742 - Physics 741 Graduate Quantum Mechanics 1...

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