I just paused writing an article where I tried to give a quick overview of quantum mechanics.
It ended up being a few paragraphs, a few images, and a crap load of links to Wikipedia. I’m not going to write anything like that anytime soon, and definitely not in a single blog post. I’m going to stick with summaries of the projects I develop and slowly work my way up to wordier subjects. Writing is the hardest part for me, so if you don’t have a bit of background in quantum mechanics, this is going to go over your head a little bit.
Like I’ve said many times in the past, I’ve looked at and derived these equations many times. Writing these apps has helped me understand them better. Here is what I’ve learned from the orbital viewer.
The wavefunctions with m=0 have a phase which is constant in space. This is pretty common in one dimensional bound states like the infinite square well or the harmonic oscillator. Because of that, I didn’t realize how weird that is. These states correspond to electrons that are frozen in space. It’s like the quantum uncertainty of the electron is completely balanced out by the compressive electrostatic force almost like little electron sausages. There is no classical correspondence to these states. There are no stationary planetary orbits.
The states with positive or negative m values are closer to classical circular orbits. The complex exponential factor adds a constant velocity around the axis. For a given energy slower electrons lie closer to the axis and are more spread out along it. As the rotation increases, the electron moves further out and becomes more concentrated on the plane orthogonal to the axis. This creates a series of stacked doughnuts.
As the energy increases for the same angular momentum, inner currents are added with alternating phases. There are a number of different ways to see this. In the full view these are added as nested doughnuts. In the slice view with the intensity cranked up, these show up as inner circles, and the nested doughnuts show up as pie shaped wedges.
This only shows up in rotational mode. The standing mode has nodes around the axis of rotation, which makes the situation visually complicated. Unfortunately many images of these orbitals show the standing waves. Using the complex exponential factor for the rotation instead of the individual elements has been a boon.
I’d never really considered what went into that factor. The complex exponential represents a completely spread out constant velocity motion. I’d thought about it a little as that’s the basic description of a quantum plane wave solution, but I think that’s a post for another day.