What is the geometry of DeBroglie standing waves?
Toroidal. Or should I say the topology is toroidal whilst the geometry is spherical. See Is the electron a photon with
toroidal topology?. It was published in the Annales de la Fondation Louis de Broglie, Volume 22, no.2, 133 (1997).
I asked a similar question here . But never received a complete answer. So I've made the question more specific to DeBroglie waves. So from what I've read DeBroglie waves are indeed standing waves within the atom, essentially storing energy in the state of a resonance.
That's right. But you should simplify matters by thinking in terms of a free electron. See the Wikipedia atomic orbitals article. It says "electrons exist as standing waves". Kick an electron out of the atom, and it still exists as a standing wave. We don't call it the wave nature of matter for nothing. Don't forget that we can make an electron and a positron out of gamma photons in gamma-gamma pair production, we can diffract and refract electrons, and then when we annihilation the electron and positron, what we get is gamma photons again.
But is there a geometry that sustains this resonance in a similar manner that a standing sound wave in a tube is sustained? Textbooks, articles illustrate the waves in a circular pattern to emphasize the quantum connection; that only specific frequencies will 'fit'. But I suspect the geometry at least is spherical if not more complex.
It's toroidal. But we're talking about a twisted spin ½ torus. See the blue torus on the Edinburgh TQFT web page. Only it's a torus that's so "fat" it has a spherical symmetry.
Can someone please provide a somewhat simple explanation, but more complete than the circle illustration?
Sure. Think of displacement current literally. Imagine you're down the beach playing in the surf. A big wave comes along and you make a little wave with your hand that scoots up and over the big wave. You might think there's no interaction but the little wave was displaced. It changed direction. Imagine you can do that to a 511keV photon so that it gets displaced into itself in the guise of a 720° double loop, and then continually displaces itself. It's now in a "closed" path rather than an "open" path.
To get the gist of it start off with the spinor depiction and think of it as something like a flat twisted inner tube. Now mentally inflate it to a torus, then keep on inflating to a horn torus then a spindle-sphere torus. That then matches the S-orbital:
See Adrian Rossiter's torus animations and the S-orbital © Copyright 2010 Encyclopaedia Britannica. . But note that there are no actual surfaces. We are talking about a wave going around and around, that's all. It's a wave in space, and space doesn't have a surface like the ocean. See Hans Ohanian's What is Spin? Feynman thought it was absurd. It isn't. What's absurd is thinking the electron is a point-particle. Note what Schrödinger said on page 27 of his 1926 paper quantization as a problem of proper values, part II. About light rays influencing one another and showing remarkable curvature.