Asymptomatic series in quantum mechanics

Abstract

In the late 19th century the quantum theory was born. Physicists like Planck and Einstein introduced the idea of quantisation, which was later formalised and justified by other scientists like Heisenberg, Schrödinger and Dirac. This new theory explained the non-intuitive results that were obtained from experiments at the time, like the famous Stern-Gerlach experiment. Arguments based on Newton’s classical mechanics were not able to describe them, and quantum mechanics was the theory created to explain them. The application of this theory to fields led to the development of quantum field theories, which are the ones physicists use at present. Here we present two formulations of quantum mechanics. An axiomatic one, where we will compute an elemental example and motivate this thesis, and the Feynman formulation, which will be useful for our calculations. We have just given a brief introduction to quantum mechanics. Chapter 2 deals with the study of a 0+0 quantum field theory, as a first example where we can compute everything explicitly. We introduce there the steepest descent method for approximating integrals and the formalism for asymptotic series. We also introduce the Borel sum, as a method to sum some divergent series. In chapter 3 we study the anharmonic oscillator. First we obtain the perturbative series for the ground state energy using Mathematica (the script is included in appendix A), and then we obtain the behaviour at large order of the coefficients using instantons. Chapter 4 summarises the work, points out the limitations and hints at future related lectures.