26.1 Integral Equations for Quantum Bound States

The script begins with an introduction to the final chapter on integral equations in quantum mechanics, emphasizing the significance of understanding both electron integral equations and their applications in quantum mechanics. The chapter is divided into two parts: the first covers bound states, which is fundamental, and the second, scattering, which is more advanced and optional. The lecturer explains the concept of effective non-local potentials in many-body systems and introduces the Schrödinger equation with a non-local potential, highlighting the complexity of solving for bound states in quantum mechanics.

This paragraph delves into the specifics of solving bound state problems in quantum mechanics, particularly in momentum space. The lecturer discusses the theoretical framework based on the Schrödinger equation in momentum space, contrasting it with the coordinate space approach. The importance of dealing with many-body systems and the use of momentum space for complex problems are highlighted. The paragraph also covers the process of converting the integral differential equation into a simpler integral equation in momentum space and introduces the concept of potential and wave function transformations to momentum space.

The script explains the numerical approach to solving integral equations by converting them into a set of simultaneous linear equations. The use of Gaussian quadrature for integration is recommended for its reliability. The process involves replacing the integral with a sum over integration points, leading to an algebraic equation that can be solved using matrix techniques. The讲师 discusses the challenges of solving these equations, including the unknown wave function and energy, and the need to solve for these at specific grid points.

This section describes the transformation of the Schrödinger equation into a matrix form, facilitating the solution of the integral equation using linear algebra techniques. The Hamiltonian matrix is defined, and the讲师 explains how the wave function and energy can be found by solving this matrix equation. The paragraph also introduces the concept of eigenvalue problems and how they relate to finding the energy levels of the quantum system.

The lecturer discusses the bound state condition, which is necessary for the existence of non-trivial solutions to the Schrödinger equation. This condition is met when the determinant of the Hamiltonian matrix minus the energy is zero. The paragraph explores the mathematical implications of this condition and how it can be used to solve for the eigenvalues, which represent the energy levels of the system.

The script introduces a specific model potential, the delta shell potential, which is used for illustrative purposes due to its analytical solutions for both bound states and scattering. The delta shell potential is characterized by an infinitely strong interaction at a specific distance, and the lecturer discusses its properties and the challenges it presents in numerical calculations.

The lecturer provides guidance on implementing numerical solutions for the delta shell potential problem. The use of matrix algebra routines is suggested, and the讲师 outlines the steps for setting up the Hamiltonian matrix, including potential and kinetic energy terms. The paragraph discusses the process of finding eigenvalues and eigenvectors, which correspond to the energy levels and wave functions of the system.

This section advises on how to analyze the results obtained from the numerical solutions, emphasizing the importance of estimating the precision of the eigenvalues. The讲师 suggests using the stability of the eigenvalues as the number of grid points increases as a measure of their validity. The paragraph also discusses the potential for numerical artifacts and the need to differentiate between true eigenvalues and noise.

The final paragraph encourages exploring the momentum space wave functions to gain an intuitive understanding of bound states in quantum mechanics. The讲师 outlines the process for determining the momentum space wave function and transforming it back to coordinate space for analysis. The importance of normalization and the expected behavior of wave functions in both spaces are discussed, with a specific focus on the delta shell potential's wave function characteristics.