Music theory and quantum mechanics

In high school science class, you probably saw a picture of an atom that looked like this:

The picture shows a stylized nucleus with red protons and blue neutrons, surrounded by three grey electrons. It’s an attractive and iconic image. It makes a nice logo. Unfortunately, it’s also totally wrong. There’s an extent to which subatomic particles are like little marbles, but it’s a limited extent. Electrons do move around the nucleus, but they don’t do it in elliptical paths as if they’re little moons orbiting a planet. The true nature of electrons in atoms is way weirder and cooler.

Pictures are a terrible way to understand the nature of quantum particles. Music theory is much better.

Quantum particles are waves

The problem with textbook images like the one above is that they mislead you into thinking of particles as “things.” Particles aren’t things. They pop in and out of being in a rapid, flickery way that’s more like the way we think of energy. What we call “particles” are really just knots or bundles of energy fields.

Protons and electrons pull on each other the way refrigerators and magnets do. If electrons really were like little moons orbiting a planet, it seems like they could orbit at any distance, and could easily fall into the nucleus to collide with the protons. And yet, this never happens. Electrons always organize themselves into very specific spatial arrangements around the nucleus. This fact was totally mysterious until scientists started conceiving of electrons as probability waves in an energy field.

You can get a good idea of how particles really behave by looking at television static, which consists of huge numbers of electrons being fired at the screen at random. Now try to imagine “static” surrounding the nucleus of an atom, and you’ll get a much better picture of what’s going on than you get from imagining moons orbiting a planet.

When electrons are in orbit around an atom or molecule, their pattern of static isn’t random the way it is in TV static. When electrons orbit atoms, their energy fields are organized into patterns of overlapping ripples. You can explore these patterns with Paul Falstad’s interactive visualizations of the subatomic world — scroll down to the Quantum Mechanics sections for his simulated hydrogen atom. The colorful blobs show the probability of electrons being found in a particular place.

Quantum harmonic oscillator

So what does this have to do with music theory? The electron field’s vibrations around an atom behave like harmonic oscillators. Electrons have harmonics, just like guitar strings do. Electron harmonics are three-dimensional instead of the one-dimensional harmonics of strings, but the underlying math is the same. These harmonics determine the arrangement and interactions of the electron wave, the same way that harmonics of a string form the basis of chords and scales. The electron field’s harmonics are called orbitals.

The physical world is made of electron harmonics

This screenshot of Falstad’s quantum harmonic oscillator applet shows the first harmonic of the electron field around an H2 molecule, two hydrogen atoms, each with one proton and one electron. This is the electron equivalent of the twelfth fret harmonic on a guitar string.

Orbitals in a hydrogen molecule

The blue blob represents the position of one electron, and the red blob is the other. At higher energy levels, the orbitals take on more complex shapes. There’s a direct analogy here to the more complex musical intervals that come from the higher harmonics in a guitar string.

Higher harmonics of the quantum harmonic oscillator

You can think of the orbitals as a structure of cubbyholes, each of which can be occupied by one electron. The cubbyholes come in pairs, and electrons “prefer” to live in filled pairs of cubbyholes. All of the structure of objects and chemistry in the world arises from the way that atoms’ outer orbitals interact. If an atom’s outermost cubbyholes are unfilled, electrons from other atoms with unfilled orbitals can fill them, locking the atoms together into molecules. All solid and liquid materials are held together by this sharing of electrons between orbitals.

Here’s the molecular structure of ice, as rendered by Masakazu Matsumoto. The red balls are oxygen atoms. The blue ones are hydrogen atoms. The yellow rods represent the bonds caused by electrons shared between the oxygen and hydrogen atoms’ outermost orbitals.

The “sixness” of ice’s structure emerges from the way that hydrogen and oxygen orbitals combine to make open slots in groups of six. You can see the “sixness” repeated up at the macroscopic scale in the shape of snowflakes and frost.

If you raise the ice’s temperature to the melting point, what you’re really doing is shooting photons at the ice, knocking the electrons out of their orbitals so they can skip more freely from atom to atom. The atoms still stick together, but not as tightly, and not in so rigid an arrangement:

If you zap even more photons into the water, you can sever the bonds between the molecules completely, freeing them to bounce around independently in the state we perceive as steam. If you zap even more photons at the steam, you can rip the molecules apart and tear the electrons from the nuclei to form plasma. Even more energy will rip the nuclei into protons and neutrons, and ridiculously more energy will rip the protons and neutrons into their constituent up and down quarks. The quarks, protons, neutrons, nuclei, atoms and molecules are all vibrating energy fields with waveforms and harmonics of their own.

Whenever I’m bored, I like to try to imagine everything around me, all the matter and energy, as resonating energy fields, cohering the way pitches cohere into chords. Who says science isn’t fun?

Teaching science with music

Albert Einstein told interviewers that he often “thought in terms of musical architectures.” Einstein was an enthusiastic amateur violinist, and an early architect of quantum mechanics. These two facts are probably related.

Did Einstein make an explicit connection between musical harmonics and quantum harmonics? Maybe we’ll never know, but the connection exists, and future scientists can benefit from it. The concept of electron orbitals is really hard. When I was in high school, my (excellent) chemistry teacher told us not to even bother trying to visualize the true nature of electrons. She was right to not try to condescend to us or mislead us, but she gave up too easily. True, she didn’t have cool interactive computer visualizations, but the school did have a great music department. If I ever get a chance to teach chemistry, first I’m going to make sure the kids get some hands-on experience with harmonics. I’ll have them experience the way that it takes more energy to produce higher harmonics, and the way those higher harmonics produce more complex musical sounds. Then we’ll go back to chemistry class and I’ll bet the kids will have an easier time.

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