2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 1,Lecture 11: Scattering theory II: Potential Scattering,Greens function formalism of Lippmann-Schwinger equation Scattering amplitude, cross section, & optical theorem Born approximation Spherical potential, Phase shift, & Partial wave method,Reading materials: Chapter 19 of Shankars PQM Chapter 7 of Sakurais MQM,Keyword: Phase shift,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 2,In the real space representation:,Lippmann-Schwinger equation,Here the causal Greens function is introduced, with definition,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 3,Physical meaning of Greens function,The propagation of the particle,Expanding by the eigen states:,The Fourier transformation is,The infinitesimal imaginary number is introduced to account for the causality: The scattered wave is determined only by the particles incoming in the past but the incoming particles in the future do not contribute.,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 4,The summation over the eigen modes can be made as an integral,The integral can be written in the spherical coordinates with the z-axis chosen along the position vector,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 5,The function to be integrated has two poles on the complex plane.,Use the residue theorem for complex function integral.,As r0, the integral has to upper contour to be converged,Physical meaning: A wavepacket propagating outward. The denominator r takes into account the increasing size of the wave front with the radius r.,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 6,Lippman-Schwinger equation,Inserting the result of the Greens function:,Physical meaning: The scattered state is the free propagation of the wave (including the free part and the scattered part) interacting with the scattering center at a earlier time (The causal Greens function is a delayed one). The propagator contains only outgoing part.,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 7,Scattering amplitude,We are interested in the wavefunction at the far field.,The distance from a position with non-zero potential to a far-field position can be expanded as,Keeping the lowest order,We can already see that the effect of the scattering potential on the far-field wave function is essentially a phase shift.,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 8,Defining the scattering amplitude (not transition amplitude),The scattering eigen state is expressed as,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 9,Cross section,The differential cross section has been calculated to be,Using the scattering amplitude:,The density of states along the scattering direction is,So the differential cross section in terms of the scattering amplitude is,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 10,Optical theorem,In terms of the scattering amplitude:,For a direct proof from the Lippmann-Schwinger equation, you are referred to L. J. Sham Lecture Notes.,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 11,Born approximation,The scattering amplitude can be calculated in orders of the scattering potential V.,e.g., the 1st order approximation is,This approximation is called the 1st order Born approximation.,e.g., in scattering by a lattice (Bragg diffraction), the quasi-momentum is conserved due to the F.T. form.,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 12,The 2nd order term is,Similarly, the n-th order Born approximation term is,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 13,Spherically symmetric potential,A special case of interest is the scattering by a spherically symmetric potential.,The scattering state of course can be expanded by the spherical harmonics times a radial part:,The radial part satisfies the radial part of the Schrdinger equation in the spherical coordinates:,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 14,Short-range potential and far-field solution,We are interested in the far-field solution.,If the potential drops with distance faster than the inverse quadratic term, we can neglect it for a distance large enough. Such kind of potential is called short-ranged potential.,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 15,The general solution at the far-field is,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 16,The asymptotical values of the Bessel functions at the far-field are sinusoidal with phase shifts:,So the asymptotical general solution can be written as,2009-10,PHY5410 Quantum Mechanics II/ Scattering theory II,Page 17,The phase shift has to be due to the potential scattering.,For a free-particle, the general solution should apply in the whole space, so the Bessel function of the 2nd kind should not appear.,So the phase shift is zero.,Actually, we will see below that in a spherically symmetric potential, the scattering amplitude is determined by the phase shift (for 2D, the potential should be rotationally symmetric, and for 1D it should have even-parity).,