ScatteringTheory - Scattering Theory Michael Fowler...

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Scattering Theory Michael Fowler 1/16/08 References: Baym, Lectures on Quantum Mechanics , Chapter 9. Sakurai, Modern Quantum Mechanics , Chapter 7. Shankar, Principles of Quantum Mechanics , Chapter 19. Introduction Almost everything we know about nuclei and elementary particles has been discovered in scattering experiments, from Rutherford’s surprise at finding that atoms have their mass and positive charge concentrated in almost point-like nuclei, to the more recent discoveries, on a far smaller length scale, that protons and neutrons are themselves made up of apparently point-like quarks. The simplest model of a scattering experiment is given by solving Schrödinger’s equation for a plane wave impinging on a localized potential. A potential V ( r ) might represent what a fast electron encounters on striking an atom, or an alpha particle a nucleus. Obviously, representing any such system by a potential is only a beginning, but in certain energy ranges it is quite reasonable, and we have to start somewhere! The basic scenario is to shoot in a stream of particles, all at the same energy, and detect how many are deflected into a battery of detectors which measure angles of deflection. We assume all the ingoing particles are represented by wavepackets of the same shape and size, so we should solve Schrödinger’s time-dependent equation for such a wave packet and find the probability amplitudes for outgoing waves in different directions at some later time after scattering has taken place. But we adopt a simpler approach: we assume the wavepacket has a well-defined energy (and hence momentum), so it is many wavelengths long. This means that during the scattering process it looks a lot like a plane wave, and for a period of time the scattering is time independent. We assume, then, that the problem is well approximated by solving the time- independent Schrödinger equation with an ingoing plane wave. This is much easier! All we can detect are outgoing waves far outside the region of scattering. For an ingoing plane wave , the wavefunction far away from the scattering region must have the form ik r e . G G ( ) ( , ) ik r ik r k e r e f r ψ θ ϕ = + . G G G G where θ , φ are measured with respect to the ingoing direction. Note that the scattering amplitude ( ) , f θ ϕ has the dimensions of length. We don’t worry about overall normalization, because what is relevant is the fraction of the incoming beam scattered in a particular direction, or, to be more precise, into a small solid angle in the direction θ , φ . The ingoing particle current (with the above normalization) is d Ω
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