Greens functions and scattering

Suppose that we have no potentials in our system, but just free electrons. Then the Hamiltonian is simply Ho = —V2, and the Schrodinger equation can be solved exactly in terms of the free-particle propagator, or Green s function, Go(r, t r, t ), which satisfies the equation  

It can be seen that the Green s function connects the value of a wave function at point r and time t to its value at a point r and time t. In fact, we can get the wave function at time t from the wave function at a time t = 0 as [39]  

This means that, just as mentioned above about Huygen s principle, every point where the wave function is non-zero at time t = 0 serves as a source for the wave function at a later time. Eq.(3.2) thus describes the propagation of the wave in free space between two different points and times, thereby giving the Green s function its second name the propagator. But what happens if we introduce a perturbing potential We assume that the potential is a point-scatterer (that is, it has no extension) and that it is constant with respect to time, and denote it by 1, which is a measure of its strength. If we now also assumes that we only have one scatterer, located at iq, the time development of the wave becomes  

We can see what happens Either the wave propagates directly to r without impinging on the scatterer (the first term), or propagates freely to the scatterer, scatters, and then propagates freely to r. 

It is easy to introduce another scatterer, i2, and write down the propagation in that case We will have one term without any scattering, now two terms with 

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