Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations

TL;DR

This paper introduces a new intuitive approach to quantum mechanics using guided numerical approximations based on path integrals, enabling the handling of complex problems like anharmonic potentials and tunneling effects.

Contribution

It presents a novel guided numerical approximation scheme that leverages path integral insights to better understand and compute quantum states in complex systems.

Findings

Successfully modeled Gaussian wave-packet evolution in anharmonic potentials

Demonstrated the method's ability to handle tunneling and instanton effects

Provided a more intuitive framework for quantum state analysis

Abstract

It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I refer to this method as a guided numerical approximation scheme and it is based upon a new look at what the path integral tells us about states in Hilbert space. I begin with simple exactly solvable models and show how to handle problems which cannot be dealt with analytically, this includes the treatment of the evolution of a Gaussian wave-packet in an anharmonic potential as well tunneling problems (i.e., instanton effects)