More interesting ideas have come up in the last few days than I have time to pursue. I'm capturing them here with a brief synopsis of each and a few pointers to documents so you can read more about it and I'll remember where to find it later.
We attended the +American Physical Society TX Section meeting last weekend in at Tarleton State Univesrity in Stephenville, TX. I say we because it was my wife, the PhD physicist in the family, my brother-in-law, a physics grad. student and me. I great time was had by all.
I got to see a second presentation of the new approach to the Schrodinger equation presentation given by Dr.Schlich at Texas A&M last week. This version was given by Dr. Scully of A&M. During the talk, Dr. Scully mentioned two things that caught my ear as I was preparing the slides from my talk, (apologies Dr. Scully.) They were, the Bohmian potential and a paper where he mentioned they had solved the hydrogen molecule energy levels very accurately using high school algebra.
This is the first I've heard of this. Apparently it all started when de Broglie came up with the concept of pilot waves. The idea as posited by de Broglie was that the electron was still a particle but that it was guided in its trajectory by the probability waves, (pilot waves) of quantum mechanics. Interestingly, when I started looking into it, in addition to the off-handed remarks about it from both Schlich and Scully, I found a recent article about the idea in the American Journal of Physics and arXiv. The article shows how the concept can be developed using the idea of wave packets following pilot waves, (picture 1 ).
Hydrogen Molecule Solution Using Algebra
It turns out the first author on this paper is another researcher in the departmetn here at A&M, so I hope to know way more about it soon! For now, while I'd hoped to be able to use it on this weeks QM homework, I didn't quite get there. This is a bit scattered, but Anatoly, (the author of the paper), recently gave a talk at the Texas Academy of Sciences meeting where he mentioned solving copuled harmonic oscillators in respect to a laser problem. I think he used ideas that are ilucidated in this article form the AJP.
Cool Connections: Quaternions, The Dirac Equation, Helicity, Weyl, Neutrinos and More
1 During Schlich's talk on , he mentioned something to the effect of "and yes, here's where Weyl predicted neutrinos."
2. David Hestenes came up indirectly at the TX SEction meeting. He helped to author the force concept inventory, one of the mainstays of physics education research. Hiis perhaps less-known accomplishment is his work on geometric algebra and spacetime algebra which each of which can show how it's easy to get to most of physics starting from quaternions.
3. We're studying Dirac's equation in quantum mechanics this week. The Dirac and Pauli matrices are both related to quaternions. Helicity came up in relation to Dirac's equation.
William Straub wrote an excellent article detailing both how quaternions lead to the Pauli and Dirac matrices and how Weyl predicted the neutrino using Dirac's equation a decade or so ahead of everyone else in 1929.
Accidental Degeneracy and Special Relativity
After my post yesterday about accidental degeneracies in the hydrogen atom, +John Baez pointed me to a document]9] he'd written that has a plethora of additional information including how the whole thing is related to the group of Lorentz rotations.
1. Scully and Schlich at PNAS
Schleich W.P., Greenberger D.M., Kobe D.H. & Scully M.O. (2013). Schrodinger equation revisited, Proceedings of the National Academy of Sciences, 110 (14) 5374-5379. DOI: 10.1073/pnas.1302475110
2. Reference to Scully's TX Section talk
3. Pilot Wave article from AJP and Arxiv
Norsen T. (2013). The pilot-wave perspective on quantum scattering and tunneling, American Journal of Physics, 81 (4) 258. DOI: 10.1119/1.4792375
4. Svidzinsky et al. on the hydrogen molecule
Svidzinsky A., Scully M. & Herschbach D. (2005). Simple and Surprisingly Accurate Approach to the Chemical Bond Obtained from Dimensional Scaling, Physical Review Letters, 95 (8) DOI: 10.1103/PhysRevLett.95.080401
5. Article on coupled harmonics oscillators
Bhattacharya M., Shi H. & Preble S. (2013). Coupled second-quantized oscillators, American Journal of Physics, 81 (4) 267. DOI: 10.1119/1.4792696
6. Hestenes on the force concept inventory
7. Hestenes on spacetime algebra
8. Straub on quaternions and Weyl
9. +John Baez on accidental degeneracies