Tag Archives: exponential function

November 10, 2022

QM 101: Diffusion in 2D

Last time I began exploring the similarity between the Schrödinger equation and a classical heat diffusion equation. In both cases, valid solutions push the high curvature parts of their respective functions towards flatness. The effect is generally an averaging out in whatever space the function occupies.

Both equations involve partial derivatives, and I ignored that in our simple one-dimensional case. Regular derivatives were sufficient. But math in two dimensions, let alone in three, requires partial derivatives.

Which were yet another hill I faced trying to understand physics math. If they are as opaque to you as they were to me, read on…

10 Comments | tags: derivatives, exponential function, partial derivatives, QM101, Schrödinger Equation | posted in Math, Physics

November 8, 2022

QM 101: Heat Diffusion

By Wyrd Smythe

This is the first of a series of posts exploring the mysterious Schrödinger Equation — a central player of quantum mechanics. Previous QM-101 posts have covered important foundational topics. Now it’s time to begin exploring that infamous, and perhaps intimidating, equation.

We’ll start with something similar, a classical equation that, among other things, governs how heat diffuses through a material. For simplicity, we’ll first consider a one-dimensional example — a thin metal rod. (Not truly one-dimensional, but reasonably close.)

Traveller’s Advisory: Math and graphs ahead!

8 Comments | tags: derivatives, exponential function, QM101, quantum, Schrödinger Equation | posted in Math, Physics

December 3, 2020

Sideband #70: The exp Function

By Wyrd Smythe

Converging…

Back in October I published two posts involving the ubiquitous exponential function. [see: Circular Math and Fourier Geometry] The posts were primarily about Fourier transforms, but the exponential function is a key aspect of how they work.

We write it as e^x or as exp(x) — those are equivalent forms. The latter has a formal definition that allows for the complex numbers necessary in physics. That definition is of a series that converges on an answer of increasing accuracy.

As a sidebar, I thought I’d illustrate that convergence. There’s an interesting non-linear aspect to it.

9 Comments | tags: exponential function, transcendental numbers | posted in Math, Sideband

October 18, 2020

Fourier Geometry

By Wyrd Smythe

Last time I opened with basic exponentiation and raised it to the idea of complex exponents (which may, or may not, have been surprising to you). I also began exploring the ubiquitous exp function, which enables the complex math needed to deal with such exponents.

The exp(x) function, which is the same as e^x, appears widely throughout physics. The complex version, exp(ix), is especially common in wave-based physics (such as optics, sound, and quantum mechanics). It’s instrumental in the Fourier transform.

Which in turn is as instrumental to mathematicians and physicists as a hammer is to carpenters and pianos.

8 Comments | tags: complex numbers, complex plane, exponential function, exponentiation, Fourier transform, Heisenberg Uncertainty | posted in Math, Physics

October 17, 2020

Circular Math

By Wyrd Smythe

Five years ago today I posted, Beautiful Math, which is about Euler’s Identity. In the first part of that post I explored why the Identity is so exquisitely beautiful (to mathematicians, anyway). In the second part, I showed that the Identity is a special case of Euler’s Formula, which relates trigonometry to the complex plane.

Since then I’ve learned how naïve that post was! It wasn’t wrong, but the relationship expressed in Euler’s Formula is fundamental and ubiquitous in science and engineering. It’s particularly important in quantum physics with regard to the infamous Schrödinger equation, but it shows up in many wave-based contexts.

It all hinges on the complex unit circle and the exp(i×π×a) function.

12 Comments | tags: 3Blue1Brown, complex numbers, complex plane, Euler's Formula, Euler's Identity, exponential function, Fourier transform | posted in Math

October 17, 2015

Beautiful Math

By Wyrd Smythe

Take a moment to gaze at Euler’s Identity:

Eulers Identity

It has been called “exquisite” and likened to a “Shakespearean sonnet.” It has earned the titles “the most famous” and “the most beautiful” formula in all of mathematics, and, in a mere seven symbols, symbolizes much of its foundation.

Today we’re going to graze on it!

14 Comments | tags: complex numbers, Euler's Formula, Euler's Identity, exponential function, Leonhard Euler, trigonometry, Yin and Yang | posted in Math, Opinion, Philosophy