Fundamental Equation Of Quantum Physics Also Describes Rings And Disks In Space

Physicists’ main goal is to be able to predict what will happen in the future based on patterns they have already observed, whether in massive systems of stars or tiny groups of atoms. Predicting changes over time usually requires developing new mathematical equations. But a researcher at California Institute of Technology recently discovered that a well-known formula, Schrödinger’s equation, governs two vastly different things: Particles smaller than an atom and the disks of matter that fill the universe.



Illustration: James Tuttle Keane (CalTech)

Familiarise yourself with Schrödinger’s equation if you like physics, since it’s one of the most important equations around. It’s the basic rule of quantum mechanics, the same way that Newton’s Laws are the basic rules of high-school physics, like throwing a baseball in the air. The difference between a particle and a baseball, though, is that particles act like single spots and waves at the same time, so the way their positions and energies change over time is different, too. While preparing for class one night, CalTech astrophysicist Konstantin Batygin realised disks of dust in space, from Saturn’s rings to the disks that form planets, also follow a specially tuned version of Schrödinger’s equation.

“I thought, OK, I want to explain what happens to astrophysical disks. How do they evolve? I realised I didn’t really know and that there wasn’t a mathematically simple explanation,” Batygin told Gizmodo. “I said, I’ll just figure it out. When the morning came, I realised it was Schrödinger’s equation that governed this whole thing. I was generally astonished.”

Batygin was wondering how ripples move through simplified versions of the disks of dust in space held together by gravity, like the disks of dust around new solar systems that will one day form into planets. He looked at this problem by dividing the disk into an infinite number of single rings of particles around the centre and calculated how small jostles, called “perturbations”, would affect the rings. Then, he combined the concentric circles, smearing all of their equations into a single equation that described the whole system. The outcome was something with the same basic mathematical form as Schrödinger’s equation.

Here’s Schrödinger’s equation for a single particle in quantum mechanics, where ψ(x,t) is the particle’s wave function, or a list of its given properties:

And here’s what Batygin calculated for how of waves would behave in astrophysical disks, where η is the initial knock on the disk.

These equations may look very different to you, but their most fundamental parts are the same. The square root of negative one (i), times a number, times the derivative of a function with respect to time, is equal to the negative square of the number times the second derivative of the function with respect to spatial coordinate x for particles, or density ρ for waves in the astrophysical disk.

Some of the specific numbers differ, since quantum mechanics describes how particles evolve based on their position and energy, while Batygin’s equation describes how perturbations evolve based on a disk system’s angular momentum and density. But the two share a deep connection. It’s like the way you can sort of tell the plot of West Side Story if you’ve seen Romeo and Juliet, even though the characters and settings are different.

If you’d like to dig deeper into the hard maths, the paper is published today in the Monthly Notices of the Royal Astronomical Society.

That is of fundamental importance, Yale astronomy professor Greg Laughlin, who was not involved in the paper, told Gizmodo. “The identification of the disk phenomena that can be described by Schrödinger’s equation means we have a lot of insight about it,” he said. “You get all the lore and understanding, now immediately applicable to a new physical situation.”

He compared it to a spring versus an electrical circuit containing an inductor and a capacitor. These are two completely different systems that follow the same mathematical equation: The behaviour of electrical current through the circuit can be described similarly to the behaviour of a weight attached to a spring moving back and forth.

There are limitations here. Batygin’s equation requires some approximations and simplifications. “If the disk is going crazy and looks like it’s on the verge of getting ripped apart, that’s not going to be covered.” It also doesn’t take into account more specific things going on in the disk, like the interactions between a pair of rocks knocking against each other. Also, no, this is not the fundamental link between general relativity and quantum mechanics that particle physicists are hunting for.

A drawing of an astrophysical disk, some maths, some other things governed by Schrödinger's equation, and a cat (Illustration: James Tuttle Keane, Caltech)

A drawing of an astrophysical disk, some maths, some other things governed by Schrödinger’s equation, and a cat. Illustration: James Tuttle Keane (Caltech)

Laughlin also pointed out that the paper is best suited for middle-aged disks around stars that haven’t formed planets yet. The Milky Way is a disk, but has spun relatively a few times in its history, so its behaviour will be more chaotic. Saturn also has its rings, but since they’re smaller, they have orbited the planet way more times than the Milky Way has spun in the same period if time, so things are pretty stable. Something in the middle, like dust around a star that hasn’t formed a planet yet, would perhaps be the best place to apply the equation.

Still, these are the kinds of disks where scientists don’t have a lot of useful mathematical tools, according to the paper. And Batygin may have opened up a whole new area of study. “It’s guaranteed to cause a lot of excitement, and this model is going to be carefully studied,” said Laughlin. “It potentially has an important new utility and a new way of getting at how planetary formation occurs.”

Andreea Font, Senior Lecturer at the Astrophysics Research Institute (ARI) at Liverpool John Moores University in the UK, pointed out that interestingly, “this is not the first time when the Schrodinger equation appears in disguise in the context of self-gravitating disks,” but this experiment takes the analogy to the Schrodinger equation in particles even further. She thought it was an “elegant solution to the long-standing problem” of evolution in disk systems like these. She agreed that the results seemed limited to special cases of these disks, and said that ultimately, comparing the predictions from this maths to observations will tell us more about how applicable the findings are.

The fundamental connectedness of the universe is definitely neat, too. Batygin was most excited about getting a deeper understanding of the mysterious equation that describe the way the most fundamental particles move.

“I remember when Schrödinger’s equation was first presented to me, I thought, where does it come from? This is great, but how did Schrödinger derive it? The professor said Schrödinger just made it up and it looks right,” said Batygin. But now that he’s derived the equations himself for his own system, “I was personally satisfied now that I know fundamentally where the equations came from, apart form all this astrophysical applications,” he said. “That helps me sleep at night.”

[MNRAS]