Dr. Paul Wilhelm

Author: Dr. Paul Wilhelm

  • Signal Theory, Quantum Mechanics, and General Relativity

    The Fundamental Role of the Fourier Transform

    Introduction

    Meet the unsung powerhouse of mathematical physics: The Fourier transform.

    It’s the formula that makes the graphical equalizer on your stereo work:

    A graphical equalizer indicates signal energies within frequency ranges.

    If you’ve ever dealt with audio production, engineering, physics, or mathematics, you probably already know that the Fourier transform takes a signal from the time domain to the frequency domain, and vice versa.

    Joseph Fourier (1768–1830), while working on the heat equation, developed his famous transform around 1807 (at the age of 39):

    $X(f) = \int_{-\infty}^{\infty} \ x(t) \cdot e^{-j2\pi f t} \ dt$

    Mathematically, the Fourier transform is just a bijective operator. It maps one space to another in a unique and reversible way. Yet, it’s so much more than just a formula.

    In particular, the Fourier transform also plays a fundamental role in quantum mechanics and is thus directly responsible for describing the reality we perceive. So bear with me as I break it down and show how it even connects to Einstein’s theory of relativity.

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  • How Dirac Got Away With Breaking the Rules

    The Hidden Vector Field Behind Magnetic Monopoles

    Magnetism: Do’s and don’ts

    Have you ever heard about magnetic monopoles? It’s what you thought you’d get when you break apart a bar magnet as a little kid.

    The bar magnet is a dipole (2-pole), because it has 1 north and 1 south pole. A magnetic monopole would be … well … a mono-pole, having 1 north or 1 south pole.

    However, ordinary nature doesn’t seem to like the idea of monopoles so much. In fact, James C. Maxwell (a.k.a. the godfather of electrodynamics, 1831–1879) expressed this fact mathematically more than 150 years ago. Heaviside (1850–1925), both a genius engineer and mathematician, later invented modern vector calculus and stepped in to re-express it as:

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  • The Deep Reason why the Magnetic Field is Circular

    No right-hand rule. No Maxwell. Just clean logic.

    Introduction

    As engineers and physicists, we all know that magnetic fields come in closed loops. So an image like this won’t surprise us much:

    However, the actual reason why the magnetic field appears circular like that is less well known.

    That’s why, when a friend once asked me that exact question, I couldn’t give a straight answer. But that was a few years back, and today I’ll finally give it a shot.

    […]

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