aqm课件lecture 24与时间无关方程其形式为亥姆.pdfVIP

Lecture 24 Scattering. The Born approximation. The time- t Schrödinger equation can be written as (which has the form of Helmholtz equation). If we find the function G(r) that solves the Helmholtz equation with a delta function source we could express  as an integral. Question to the class: proof that this still satisfies the Schrödinger equation. Note that what we are trying to s essentially to write Schrödinger equation in an integral form rather than solve it. The function G(r) is called the Greens function for the Helmholtz equation. Derivation (see pages 409-411 of the textbook) shows that it is We can add to it any function that satisfies the homogeneous Helmholtz equation Now we arrive to the integral form of the Schrödinger equation The First Born Approximation We suppose that scattering potential V(r ) is localized about r =0, i.e. potential drops 0 0 to zero outside of finite region. It is a typical case for a scattering problem. We would like to calculate the wave function far away from the scattering center. Therefore, we can assume for all points in our integral. Then, Lets use this approximation in Note that we need to be more careful with approximating the exponential term than the denominator. ase of scattering ne wave Then, our wave function can be written in a form So far, we only assumed Now, we use the Born approximation. Lets assume that the potential does not significantly alter the wave function (weak potential approximation). Then, Scattering amplitude in Born approximation. As before, the differential and total cross sections are given by We can now consider two cases: (1) Low-energy scattering (2) Spherically symmetric potential. Low energy scattering In the case of low energy scattering, we can consider exponential factor to be cons