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# inclass2 - Computer project 2 PHZ 5156 Results due Thursday...

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Computer project 2 PHZ 5156 Results due Thursday, September 30 Please submit your code and plots wherever requested. Results can be handed in either as a hardcopy, or as an electronic document (e.g. tex, latex, MS word, or even a .pdf) sent via email. 1. A wave function in one dimension evolves according to the Schrodinger equation as, i ∂ψ ( x, t ) ∂t = - 2 2 m 2 ∂x 2 + V ( x ) ψ ( x, t ) Show that if V ( x ) = 0 for a t = 0 wave packet given by ψ ( x, t = 0) = 1 σ 1 2 0 π 1 4 e ik 0 x e - ( x - x 0 ) 2 / 2 σ 2 0 that the analytic expression for the time evolution is given by ψ ( x, t ) = 1 ( σ 0 σ ) 1 4 π 1 4 e ik 0 ( x - st/ 2) e - ( x - x 0 - st ) 2 / 2 σ 0 σ with s = k 0 /m and σ = σ 0 + i t 0 . Simply prove that the solution works at t = 0 and then satisfies the time-dependent Schrodinger equation. This result can be found analytically also using Green functions. 2. Implement the Crank-Nicholson scheme to calculate the quantum-mechanical time evolution of a one-dimensional Gaussian wave packet with hard wall boundaries, so that the wave function vanishes at x = 0 and x = l .

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