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homework3 - Homework 3 PHZ 5156 Due Tuesday September 26 1...

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Homework 3 PHZ 5156 Due Tuesday, September 26 1. In the second computer project, we showed that we could solve the diffusion equation for a point source at t = 0. This is what is known as a Green function. Then time-dependent Schrodinger equation for a free particle in one dimension i ∂ψ ∂t = - 2 2 m 2 ψ ∂x 2 can also be solved for a t = 0 wave function ψ ( x, t = 0) = δ ( x - x ). The result is given by the free-particle propagator (a.k.a. a Green function), K ( x , x, t ) = m 2 πi t exp im ( x - x ) 2 2 t Notice the similarity of this result to the result of problem 4 on the last home- work. Given that for any t = 0 wave function ψ ( x, t = 0), we can find the time evolution from ψ ( x, t ) = -∞ dx ψ ( x , t = 0) K ( x , x, t ) Using this approach, show that for a t = 0 wave packet given by ψ ( x, t = 0) = 1 σ 0 π 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 π 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 . This is the same as the question on computer
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