Editing Instanton (section)

=== Motivation of considering instantons ===
Consider the quantum mechanics of a single particle motion inside the double-well potential <math>V(x)={1\over 4}(x^2-1)^2.</math>
The potential energy takes its minimal value at <math>x=\pm 1</math>, and these are called classical minima because the particle tends to lie in one of them in classical mechanics. There are two lowest energy states in classical mechanics.

In quantum mechanics, we solve the [[Schrödinger equation]]

:<math>-{\hbar^2\over 2m}{\partial^2\over \partial x^2}\psi(x)+V(x)\psi(x)=E\psi(x), </math>

to identify the energy eigenstates. If we do this, we will find only the unique lowest-energy state instead of two states. The ground-state wave function localizes at both of the classical minima <math>x=\pm 1</math> instead of only one of them because of the quantum interference or quantum tunneling.

Instantons are the tool to understand why this happens within the semi-classical approximation of the path-integral formulation in Euclidean time. We will first see this by using the WKB approximation that approximately computes the wave function itself, and will move on to introduce instantons by using the path integral formulation.