# A Weak Derivation of the Momentum Operator

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.

# Path Integral Free Propagator

In quantum field theory, the propagator gives a probability amplitude for a particle traveling from some point $(t_i,x_i)$ to $(t_f,x_f)$ 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

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