Different Bootstrap Matrices in Many QM Systems

Abstract: The bootstrap is a technique recently developed to get energy eigenvalues of bound states and correlation functions. There are three crucial steps - recursive equations, positivity constraints, search space. We calculate recursive equations of many representative quantum mechanics systems, such as polynomial potential, exponential potential, Yukawa potential and electromagnetic potential. Two kinds of bootstrap matrices, which are about the coordinate and coupling of the coordinate with the momentum, and their ability of constraining equations are displayed. Nextly, we analyze possible questions in numerical search, including the importance of constraints and step length, eigen-energy level and the degeneracy of energy. Finally, we try to explain why the bootstrap work well by analyzing positivity constraints of creation operator and annihilation operator in harmonic oscillator. This article summarizes most knowledge of bootstrapping quantum mechanics (QM), and displays specific bootstrap equations and bootstrap matrices of different QM systems.