leios
Inorganic chemist with ideas: elementary quantum mechanics using geometric algebra or solving diffraction patterns at home
About the author
I'm a chemistry grad student at UW-Madison and my research is on the structural chemistry of intermetallics. I'm primarily a computational chemist, but I have experience with experimental work as well. I'm intending to include a friend of mine who's also a member of my research group in this SoME entry.
I am also currently writing a package, Xtal.jl, which I use in some of my research. It's intended to provide users with a number of convenient types and methods that are commonly used in working with the results of DFT calculations on crystals.
Ideally, I'd like to make a video, but I'd also be cool with an interactive webpage. I don't have too much programming experience, but I do use Julia quite a bit and I can put together Pluto notebooks. I'd definitely like to try my hand at animation through Javis, manim, or even Blender, but I'm not very familiar with either. I might be better off playing the role of a domain expert (though I'm certainly not a star in my field or anything - more experts are welcome!)
My ideas
I actually have a few ideas for what I'd like to make a video on. What I choose to make will depend on others' interests as well, so I'm not married to the ideas that are listed here.
"A new framework for quantum chemistry: revisiting quantum mechanics with geometric algebra"
I'm a huge fan of geometric algebra as a framework for physics, which I discovered with sudgylacmoe's video, A Swift Introduction to Geometric Algebra. If you haven't seen it, give it a watch! (Geometric algebras are equivalent to real Clifford algebras)
One thing that bugs me about physical chemistry is that the math prerequisites are insufficient. The only reason I did well in quantum as an undergrad was because I watched 3Blue1Brown's linear algebra series - linear algebra is not a requirement for most chemistry students. If chemistry students are already going to be effectively relearning linear algebra while learning quantum, I think introducing them to geometric algebra instead might make more sense. However, although there are papers and books discussing the application of geometric algebra to quantum mechanics, many of them are too high level to be helpful to someone just learning quantum - we can't assume they know what Pauli matrices are, for instance.
Here I'll list several different points that seriously confuse students approaching the subject in chemistry.
- What is a Hamiltonian, exactly? We can use phase space to visualize the relationship between position and momentum, and how those quantities change over time. A particle in a harmonic oscillator, for instance, traces out a circle or ellipse in phase space. We can also extend this from a point particle to a probability distribution to understand how ensembles of particles, or a single particle with uncertainties in position and momentum, move through space.
- Why do complex numbers show up? We measure physical quantities with real numbers, and it can be surprising for students to see an equation that contains complex numbers which aren't even clearly interpretable. However, we can find good reason to use complex numbers even in classical mechanics: in order to traverse level sets of the Hamiltonian, one can take the gradient of the Hamiltonian and rotate all of the vectors in the gradient field and rotate them 90 degrees. This 90 degree rotation is naturally identified with the imaginary unit! I'm not 100% sure on this, but with the identification of i as the bivector generated by position and momentum, I think the commutation relations between position and momentum can be recovered.
- What is a spinor? What does it mean for electrons to be spin 1/2? It turns out, even when they're not explicitly introduced, students of quantum chemistry are already dealing with spinors. The giveaway is the resemblance between the inner product as defined on quantum states and the application of a rotor to a multivector in geometric algebra. In 2 dimensions, spinors are just complex numbers, which behave nicely. However, in 3 dimensions, they become very complicated, and this is where Pauli matrices and 2D complex vectors are invoked. Geometric algebra allows for the construction of 3D spinors without directly invoking complex numbers - the 2D complex vectors are isomorphic with quaternions, which comprise the even subalgebra of the 3D geometric algebra.
However, I'm not clear on the details of a few points mentioned here. For instance, in 1D the only rotors are 1 and -1: the complex numbers should not show up with a construction like this. I also am not sure about the commutation relations falling this naturally out of the identification of i as a bivector in phase space.
"Mom says we have an XRD at home: solving the shape of a pixel in a phone screen using a green laser pointer"
If you shine a green laser pointer at a cell phone screen, its reflection will show not just the original beam, but a grid of diffraction peaks generated by the pixels underneath the glass accompanying it. This can serve as a really simple home demonstration of the von Laue diffraction conditions, and we can relate that to the Fourier transform.
However, I suspect that we can use images of the diffraction pattern to work backwards and solve for the shape of a single pixel in a screen. I have taken a class in crystallography and have some familiarity with the algorithms used to solve structures and refine models, so in principle I could assist with the creation of an interactive module or video that could allow someone to take images of the diffraction pattern of a laser beam and use that to solve for the shape of a pixel in a phone or computer screen.
This one has less detail than the above, but I think it's something that could be more easily turned into a home experiment, which I am all for!
Contact details
You can email me at bsflores [at] wisc.edu, or message me on Discord: @brainandforce#2844 (you can find me in the SoME discord). If you send a friend request, please accompany it with a message indicating that you're from SoME.
