Smoke in the Leaf

In quantum chemistry this semester, we’ve been solving eigenvalue equations over and over. I think some of my students get what we’re doing, but I still have moments in office hours where a student seems to struggle with the very concept. Perhaps the narrow way in which I introduce this topic isn’t helping some of the students.

 

It was therefore refreshing to read a different take on eigenvalues in Jordan Ellenberg’s Shape, which I recently blogged about. Starting with geometric progressions, he shows examples of how one can extract a specific number that seems to control the rate of geometric growth. This number is the eigenvalue. He then introduces the Fibonacci sequence and shows that it arises from the difference of two geometric progressions, controlled by the eigenvalues -0.618 and +1.618, the latter famously known as the “golden ratio”. I like the way Ellenberg discusses the nature of eigenvalues:

 

[They] capture something deep and global about the system’s behavior. They’re not properties of any individual part of the system, but emerge from interaction among its parts. The algebraist James Joseph Sylvester called these numbers the latent roots – ‘latent,” as he vividly explained, “in a somewhat similar sense as vapour may be said to be latent in water or smoke in tobacco-leaf.” Unfortunately English-speaking mathematicians have preferred to half translate David Hilbert’s word Eigenwert, which is German for “inherent value”.

 

In my class, we talk about using Hermitian operators to extract eigenvalues from a wavefunction. I’ve tried to give my students the sense that you’re pulling out a number from some sort of global system that is captured by a mathematical (wave)function, but sometimes when I talk about math the students seem befuddled. Perhaps Ellenberg’s “smoke in the leaf” will give them a physical picture of what we’re trying to do – perhaps capturing the vapor in a GC-MS and extracting an output number, e.g., the mass of a particular molecule in the vapor.

 

Ellenberg expands the idea with more examples beyond the geometric progression “growth” of a pandemic and R₀ values. There’s an example of Google PageRank searching and Markov (random) walks. There’s a segue into Monopoly the boardgame, where apparently Illinois Ave (in the U.S. version) is where a counter would spend most of its time in the long run (i.e., you’d need to play a very, very long game to notice this) once the system approaches ergodicity. You can capture this in an eigenvector: “something inherent to the long-term behavior of a system that’s not apparent just by looking at it, something latent like the smoke in the leaf.”

 

What’s helpful at the end of the chapter: Ellenberg provides a worked example of two infinite sequences and two operations. One is a shift, where moving all the numbers in a sequence one spot to the left looks like you’ve multiplied the original sequence by a constant to get the new sequence. The other is a pitch where you multiply each term of a sequence by its position in the sequence. This leads to a discussion of eigensequences, akin to eigenfunctions, followed by demonstrating these as examples of non-commuting operations. And yes, this leads to Heisenberg’s Uncertainty Principle. My students are regularly puzzled by this even though they follow the mathematics. I don’t think I’ve done a great job helping to shed light on this, so I’ve scanned some pages of Ellenberg’s book for my students to read. Maybe it will help shed light, maybe it won’t.

 

Next week we’ll be looking at Huckel Molecular Orbital Theory for delocalized pi-systems. One of the examples will be butadiene. Interestingly, the eigenvalues for the “resonance” integral (in units of beta) are -1.618, -0.618, +0.618, +1.618 for the four molecular orbitals – seemingly related to the two eigenvalues controlling the Fibonacci sequence. I don’t know if there’s a connection between the two, or I should say that I haven’t spent the time thinking about it more carefully. My brain is somewhat fried, and I’m just happy to make it to the Thanksgiving holiday. My students are also tired, maybe more so. It’s been a grueling semester.

 

This chapter in Ellenberg’s book ends with the “notes in the chord”. It’s kinda cool. When you hear a chord triad, you can separate out the three notes and their three eigenvalues mathematically using a Fourier transform. But more amazingly, a trained ear can actually hear these notes in the chord “even if you don’t know calculus… because this deeply geometric computation… is also carried out by a curled-up piece of meat in your ear called the cochlea.” It’s amazing what these small body parts can do. An eigenvalue separator in our very ears!