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

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Unformatted text preview: 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 ) = r 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 ) = Z ∞-∞ 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 π e ik x e- ( x- x ) 2 / 2 σ 2 that the analytic expression for the time evolution is given by ψ ( x, t ) = 1 √ σ π e ik ( x- st/ 2)...
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This note was uploaded on 08/08/2011 for the course PHZ 5156 taught by Professor Johnson,m during the Fall '08 term at University of Central Florida.

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

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