Here is a cute trick to derive the momentum operator in quantum mechanics. Usually, we taken this for a given or a definition, however there are ways to prove the relationship. Here we explore the relationship between position and momentum using Fourier Transforms.

# Monthly Archives: April 2016

# Path Integral Free Propagator

In quantum field theory, the propagator gives a probability amplitude for a particle traveling from some point $latex (t_i,x_i)$ to $latex (t_f,x_f)$ with a certain energy and momentum. These propagators are the first steps into quantum field theory, I aim to bring these to the masses. The one that is most interesting and easy to grasp is the free propagator for a free field theory. The following derivation is inspired by Anthony Zee’s *Quantum Field Theory in a Nutshell* of UC Santa Barbara.

# Proof of Euler’s Formula

Around 1740, Leonard Euler discovered a formula that connected functions of complex arguments to trigonometric functions, effectively forming a link between analytic functions and geometric functions which eventually extended to topology, differential equations, and mathematical physics. All of this began with one simple formula, lauded by *Richard Feynman* as “the most remarkable formula in mathematics,” and it is

$$e^{ix}=\cos(x)+i\sin(x), \text{ with } x\in\mathbb{R}. $$