Physicistsâ main goal is to be able to predict what will happen in the future based on patterns theyâve 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, SchrĂśdingerâs equation, governs two vastly different things: particles smaller than an atom and the disks of matter that fill the universe.

Familiarize yourself with SchrĂśdingerâs equation if you like physics, since itâs one of the most important equations around. Itâs the basic rule of quantum mechanics, the same way that Newtonâ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 realized disks of dust in space, from Saturnâs rings to the disks that form planets, also follow a specially tuned version of SchrĂśdingerâs equation.

âI thought, okay, I want to explain what happens to astrophysical disks. How do they evolve? I realized I didnât really know and that there wasnât a mathematically simple explanation,â Batygin told Gizmodo. âI said, Iâll just figure it out. When the morning came, I realized it was SchrĂśdingerâs equation that governed this whole thing. I was generally astonished.â

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 center and calculated how small jostles, called âperturbations,â 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 SchrĂśdingerâs equation.

Hereâs SchrĂśdingerâs equation for a single particle in quantum mechanics, where Ď(x,t) is the particleâs wave function, or a list of its given properties:

And hereâs what Batygin calculated for how of waves would behave in astrophysical disks, where Îˇ 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 Ď 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 Batyginâs equation describes how perturbations evolve based on a disk systemâs angular momentum and density. But the two share a deep connection. Itâs like the way you can sort of tell the plot of West Side Story if youâve seen Romeo and Juliet, even though the characters and settings are different.

If youâd like to dig deeper into the hard math, 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. âThe identification of the disk phenomena that can be described by SchrĂśdingerâs equation means we have a lot of insight about it,â he said. âYou get all the lore and understanding, now immediately applicable to a new physical situation.â

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 behavior of electrical current through the circuit can be described similarly to the behavior of a weight attached to a spring moving back and forth.

There are limitations here. Batyginâs equation requires some approximations and simplifications. âIf the disk is going crazy and looks like itâs on the verge of getting ripped apart, thatâs not going to be covered.â It also doesnâ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.

Laughlin also pointed out that the paper is best suited for middle-aged disks around stars that havenât formed planets yet. The Milky Way is a disk, but has spun relatively a few times in its history, so its behavior will be more chaotic. Saturn also has its rings, but since theyâre smaller, theyâve 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 hasnât formed a planet yet, would perhaps be the best place to apply the equation.

Still, these are the kinds of disks where scientists donât have a lot of useful mathematical tools, according to the paper. And Batygin may have opened up a whole new area of study. âItâs guaranteed to cause a lot of excitement, and this model is going to be carefully studied,â said Laughlin. âIt potentially has an important new utility and a new way of getting at how planetary formation occurs.â