Bootstrap Method in Harmonic Oscillator

TL;DR

This paper investigates the bootstrap method in quantum harmonic oscillators, revealing its connection to Dirac's ladder operators and providing an analytical understanding of its effectiveness across systems.

Contribution

It demonstrates that the bootstrap method reduces to Dirac's ladder operator problem in harmonic oscillators, offering an analytical explanation for its success.

Findings

Bootstrap method reduces to Dirac's ladder operator problem.

Analytical solution explains bootstrap's effectiveness.

Supports bootstrap as a numerical analogue of Dirac's approach.

Abstract

Recently, an application of the numerical bootstrap method to quantum mechanics was proposed, and it successfully reproduces the eigenstates of various systems. However, it is unclear why this method works. In order to understand this question, we study the bootstrap method in harmonic oscillators. We find that the problem reduces to the Dirac's ladder operator problem and is exactly solvable analytically. Our result suggests that the bootstrap method may be regarded as a numerical version of the Dirac's approach and it may explain why it works in various systems.