Fundamental Equation of Quantum Physics Also Describes Rings and Disks in Space
Physicistsâ main goal is to be able to predict what will happen in the future based on patterns theyâve 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, SchrÃ¶dingerâs equation, governs two vastly different things: particles smaller than an atom and the disks of matter that fill the universe.
Familiarize yourself with SchrÃ¶dingerâs equation if you like physics, since itâs one of the most important equations around. Itâs the basic rule of quantum mechanics, the same way that Newtonâs Laws are the basic rules of high-school physics, like throwing a baseball in t he 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 Saturnâs rings to the disks that form planets, also follow a specially tuned version of SchrÃ¶dingerâs equation.
âI thought, okay, I want to explain what happens to astrophysical disks. How do they evolve? I realized I didnât really know and that there wasnât a mathematically simple explanation,â Batygin told Gizmodo. âI said, Iâll just figure it out. When the morning came, I realized it was SchrÃ¶dingerâs 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 s olar 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 âperturbations,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 SchrÃ¶dingerâs equation.
Hereâs SchrÃ¶dingerâs equation for a single particle in quantum mechanics, where Ï(x,t) is the particleâs wave function, or a list of its given properties:
And hereâs what Batygin calculated for howof waves would behave in astrophysical disks, where Î· 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 nu mber, 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 Ï 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 Batyginâs equation describes how perturbations evolve based on a disk systemâs angular momentum and density. But the two share a deep connection. Itâs like the way you can sort of tell the plot of West Side Story if youâve seen Romeo and Juliet, even though the characters and settings are different.
If youâd 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 G reg Laughlin, who was not involved in the paper, told Gizmodo. âThe identification of the disk phenomena that can be described by SchrÃ¶dingerâs equation means we have a lot of insight about it,â he said. âYou 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. Batyginâs equation requires some approximations and simplifications. âIf the disk is going crazy and looks like itâs on the verge of getting ripped apart, thatâs not going to be covered.â It also doesnât take into account more specific things going on in the disk, like the interactions b etween 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 havenât 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 theyâre smaller, theyâve 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 hasnât formed a planet yet, would perhaps be the best place to apply the eq uation.
Still, these are the kinds of disks where scientists donât have a lot of useful mathematical tools, according to the paper. And Batygin may have opened up a whole new area of study. âItâs guaranteed to cause a lot of excitement, and this model is going to be carefully studied,â said Laughlin. âIt potentially has an important new utility and a new way of getting at how planetary formation occurs.â
Andreea Font, Senior Lecturer at the Astrophysics Research Institute (ARI) at Liverpool John Moores University in the UK, pointed out that interestingly, âthis is not the first time when the Schrodinger equation appears in disguise in the context of self-gravitating disks,â but this experiment takes the analogy to the Schrodinger equation in particles even further. She thought it was an âelegant solution to the long-standing problemâ of evolution in disk systems like these. She agreed that the results seemed limited to special case s of these disks, and said that ultimately, comparing the predictions from this math 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.
âI remember when SchrÃ¶dingerâs equation was first presented to me, I thought, where does it come from? This is great, but how did SchrÃ¶dinger derive it? The professor said SchrÃ¶dinger just made it up and it looks right,â said Batygin. But now that heâs derived the equations himself for his own system, âI was personally satisfied now that I know fundamentally where the equations came from, apart form all this astrophysical applications,â he said. âThat helps me sleep at night.â
This post has been updated to include a quote from Andreea Font.