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
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
It’s been a minute or two since my last post. In large part, I’ve been on a three-week coding binge, moving some projects along and doing improvements and minor bug fixes on library code. At this point I’m a bit burned out and definitely over the binge. I don’t get the mood to really tuck in like this much anymore, so when that mood does strike, I surrender to it.
One consequence of not posting is that the longer I’m away from it the harder it seems to start up again. Some part of me finds this unrewarding and unsatisfying, but another part of me enjoys the writing and is drawn to it. Also, I still struggle with maintaining an artificial politeness rather than unleashing my inner guido (and I’m so tempted some days).
Anyway, without further ado, another edition of Friday Notes.
22 Comments | tags: derivatives, James Webb Space Telescope, JWST | posted in Friday Notes
I’ve always had a strong curiosity about how things work. My dad used to despair how I’d take things apart but rarely put them back together. My interest was inside — in understanding the mechanism. (The irony is that I began my corporate career arc as a hardware repair technician.)
My curiosity includes a love of discovery, especially unexpected ones, and extra especially ones I stumble on myself. It’s one thing to be taught a neat new thing, but a rare delight to figure it out for oneself. It’s like hitting a home run (or at least a base-clearing double).
Recently, I was delighted to discover something amazing about spheres.
9 Comments | tags: derivatives, geometry, holographic principle | posted in Math, Sideband
Today is the first Earth-Solar event of 2021 — the Vernal Equinox. It happened early in the USA: 5:37 AM on the east coast, 2:37 AM on the west coast. Here in Minnesota, it happened at 4:37 AM. It marks the first official day of Spring — time to switch from winter coats to lighter jackets!
Have you ever thought the Solstices seem more static than the Equinoxes? The Winter Solstice particularly, awaiting the sun’s return, does it seem like the change in sunrise and sunset time seems stalled?
If you have, you’re not wrong. Here’s why…
17 Comments | tags: derivatives, equinox, sine wave, Solstice, spring equinox, Summer Solstice, vernal equinox, Winter Solstice | posted in Life, Math
There are many science-minded authors and working physicists who write popular science books. While there aren’t as many math-minded authors or working mathematicians writing popular math books, it’s not a null set. I’ve explored two such authors recently: mathematician Steven Strogatz and author David Berlinski.
Strogatz wrote The Joy of X (2012), which was based on his New York Times columns popularizing mathematics. I would call that a must-read for anyone with a general interest in mathematics. I just finished his most recent, Infinite Powers (2019), and liked it even more.
Berlinski, on the other hand, I wouldn’t grant space on my bookshelf.
11 Comments | tags: calculus, David Berlinski, derivatives, fun with numbers, integrals, numbers, Steven Strogatz, The Joy of X | posted in Books, Math