Question

Homework (scattering) 1. Consider the time dependent Schrödinger equation written in the form where 0 2mo As it is well known the temporal evolution of a wave function ψ( t) known at a specific time t is uniquely determined for all future times t, > t as well as for all past times t' < t. Moreover, given that the Schrödinger equation is linear, which means that the solutions are sup times must be linear, the wave function must obey a linear integral equation of the form erposable and the relation between solutions at different where dVol is the volume element, x = (三t) and G(x,x') is the Green function containing the entire information with respect to the evolution from (x) to ψ(z) during the time from t to t. It is worth noticing that the above equation does not distinguish between the two temporal directions of evolution, t'>t and t'<t. However, we can distinguish between these situations rewriting the above equation as where Θ(t) denotes the step function and the Green functions G (z,z) allow a propagation from ψ(x) to ψ(r) only in a positive or a negative temporal direction. Due to this causal behavior, we define G+ as the retarded propagator and G the advanced p (a) Demonstrate that G satisfies propagator (b) Demonstrate that the solution of the above diffferential equation can be written as = dtdVol and Go±(z.z) corresponds to the free propagator which satisfy where d c) Demonstrate that an integral form of the the wave function which solve the Schrödinger equation, by is given t-Foo where Eim

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Answer 1

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