26.3 Integral Equations for Quantum Scattering

The speaker begins the final lecture of the book, indicating that the content is of a graduate or research level, involving complex mathematics and computing. The lecture focuses on quantum mechanics and scattering in momentum space, an area the speaker has researched for decades. The bound state problem in momentum space is reviewed as a prerequisite, and the non-local potential in a multi-particle medium is discussed. The speaker introduces the Schrödinger equation in this context and explains the transition to solving problems in momentum space, including the concept of a center of momentum system. The lecture aims to open doors to further opportunities in the field.

The speaker delves into the Lippmann-Schwinger equation, an integral form of the Schrödinger equation used for scattering problems. The equation is explained as a means to find the scattering amplitude, which is related to the experimental observable, the scattering cross-section. The challenges of solving the integral equation are discussed, including the singularity issue and the use of the Cauchy principal value integration to handle it. The speaker also explains the concept of a singular integral and its relevance to the problem at hand, emphasizing the need for both analytic and computational skills to address these challenges.

The speaker discusses methods for computing singular integrals, which are crucial for solving the scattering problem in quantum mechanics. Three different approaches are presented: giving the energy a small imaginary part to avoid the singularity, using the Cauchy principal value prescription, and employing Hilbert transforms. The speaker illustrates these methods with mathematical functions and explains how they can be applied to the Lippmann-Schwinger equation. The importance of careful numerical work when dealing with singularities is highlighted, and the speaker provides insights into how to handle these in computations.

The speaker outlines the process of converting the integral equation, specifically the Lippmann-Schwinger equation, into a matrix equation to facilitate numerical computation. The method involves rewriting the equation with a subtraction term to eliminate the singularity and then approximating the integration using Gaussian quadrature points. The resulting set of algebraic equations is then expressed in matrix form, allowing for the use of matrix libraries to solve for the unknowns. The speaker emphasizes the clever manipulation of the equation to ensure it can be computed numerically.

The speaker describes the process of solving the matrix equation obtained from the Lippmann-Schwinger equation. The equation is manipulated to isolate the unknown vector, which represents the scattering amplitudes. The solution involves matrix inversion, which is possible due to the nature of scattering problems having a continuous spectrum. The speaker provides a Python code snippet that demonstrates how to perform the matrix inversion and solve for the scattering amplitudes, which are related to the cross-section measured in experiments.

The speaker presents an analysis of the numerical results obtained from solving the scattering problem using the matrix inversion method. The results are compared with the analytic solution for the delta shell potential, a specific case in quantum mechanics. The speaker notes the challenges in obtaining precise numerical results, especially near resonance peaks, and suggests increasing the number of grid points for improved accuracy. The importance of validating numerical solutions with analytic checks is emphasized.

In the final paragraph, the speaker concludes the lecture series by encouraging students to undertake the computation for the delta shell potential in momentum space. The speaker provides guidance on how to approach the problem and what to expect in terms of results. There is an acknowledgment of the challenges in numerically integrating the delta shell potential and a reminder to compare the numerical results with the provided analytic solution. The speaker thanks the audience for their participation and expresses hope that they will continue to engage with the material.