Researchers who want to predict the behavior of systems governed by quantum mechanics—an electron in an atom, say, or a photon of light traveling through space—typically turn to the Schrödinger equation. Devised by Austrian physicist Erwin Schrödinger in 1925, it describes subatomic particles and how they may display wavelike properties such as interference. It contains the essence of all that appears strange and counterintuitive about the quantum world.
But it seems the Schrödinger equation is not confined to that realm. In a paper just published in Monthly Notices of the Royal Astronomical Society, planetary scientist Konstantin Batygin of the California Institute of Technology claims this equation can also be used to understand the emergence and behavior of self-gravitating astrophysical disks. That is, objects such as the rings of the worlds Saturn and Uranus or the halos of dust and gas that surround young stars and supply the raw material for the formation of a planetary system or even the accretion disks of debris spiraling into a black hole.
And yet there’s nothing “quantum” about these things at all. They could be anything from tiny dust grains to big chunks of rock the size of asteroids or planets. Nevertheless, Batygin says, the Schrödinger equation supplies a convenient way of calculating what shape such a disk will have, and how stable it will be against buckling or distorting. “This a fascinating approach, synthesizing very old techniques to make a brand-new analysis of a challenging problem,” says astrophysicist Duncan Forgan of the University of Saint Andrews in Scotland, who was not part of the research. “The Schrödinger equation has been so well studied for almost a century that this connection is clearly handy.”
From Classical to Quantum
This equation is so often regarded as the distilled essence of “quantumness” that it is easy to forget what it really represents. In some ways Schrödinger pulled it out of a hat when challenged to come up with a mathematical formula for French physicist Louis de Broglie’s hypothesis that quantum particles could behave like waves. Schrödinger drew on his deep knowledge of classical mechanics, and his equation in many ways resembles those used for ordinary waves. One difference is that in quantum mechanics the energies of “particle–waves” are quantized: confined to discrete values that are multiples of the so-called Planck’s constant h, first introduced by German physicist Max Planck in 1900.
This relation of the Schrödinger equation to classical waves is already revealed in the way that a variant called the nonlinear Schrödinger equation is commonly used to describe other classical wave systems—for example in optics and even in ocean waves, where it provides a mathematical picture of unusually large and robust “rogue waves.”
But the normal “quantum” version—the linear Schrödinger equation—has not previously turned up in a classical context. Batygin says it does so here because the way he sets up the problem of self-gravitating disks creates a quantity that sets a particular “scale” in the problem, much as h does in quantum systems.
Whether around a young star or a supermassive black hole, the many mutually interacting objects in a self-gravitating debris disk are complicated to describe mathematically. But Batygin uses a simplified model in which the disk’s constituents are smeared and stretched into thin “wires” that loop in concentric ellipses right around the disk. Because the wires interact with one another through gravity, they can exchange orbital angular momentum between them, rather like the transfer of movement between the gear bearings and the axle of a bicycle.
This approach uses ideas developed in the 18th century by the mathematicians Pierre-Simon Laplace and Joseph-Louis Lagrange. Laplace was one of the first to study how a rotating clump of objects can collapse into a disklike shape. In 1796 he proposed our solar system formed from a great cloud of gas and dust spinning around the young sun.
Batygin and others had used this “wire” approximation before, but he decided to look at the extreme case in which the looped wires are made thinner and thinner until they merge into a continuous disk. In that limit he found the equation describing the system is the same as Schrödinger’s, with the disk itself being described by the analog of the wave function that defines the distribution of possible positions of a quantum particle. In effect, the shape of the disk is like the wave function of a quantum particle bouncing around in a cavity with walls at the disk’s inner and outer edges.
The resulting disk has a series of vibrational “modes,” rather like resonances in a tuning fork, that might be excited by small disturbances—think of a planet-forming stellar disk nudged by a passing star or of a black hole accretion disk in which material is falling into the center unevenly. Batygin deduces the conditions under which a disk will warp in response or, conversely, will behave like a rigid body held fast by its own mutual gravity. This comes down to a matter of timescales, he says. If the angular momentum of the objects orbiting in the disk is transferred from one to another much more rapidly than the perturbation’s duration, the disk will remain rigid. “If on the other hand the self-interaction timescale is long compared with the perturbation timescale, the disk will warp,” he says.
Is “Quantumness” Really So Weird?
When he first saw the Schrödinger equation materialize out of his theoretical analysis, Batygin says he was stunned. “But in retrospect it almost seems obvious to me that it must emerge in this problem,” he adds.
What this means, though, is the Schrödinger equation can itself be derived from classical physics known since the 18th century. It doesn’t depend on “quantumness” at all—although it turns out to be applicable to that case.
That’s not as strange as it might seem. For one thing, science is full of examples of equations devised for one phenomenon turning out to apply to a totally different one, too. Equations concocted to describe a kind of chemical reaction have been applied to the modeling of crime, for example, and very recently a mathematical description of magnets was shown also to describe the fruiting patterns of trees in pistachio orchards.
But doesn’t quantum physics involve a rather uniquely odd sort of behavior? Not really. The Schrödinger equation does not so much describe what quantum particles are actually “doing,” rather it supplies a way of predicting what might be observed for systems governed by particular wavelike probability laws. In fact, other researchers have already shown the key phenomena of quantum theory emerge from a generalization of probability theory that could, too, have been in principle devised in the 18th century, before there was any inkling that tiny particles behave this way.
The advantage of his approach is its simplicity, Batygin notes. Instead of having to track all the movements of every particle in the disk using complicated computer models (so-called N-body simulations), the disk can be treated as a kind of smooth sheet that evolves over time and oscillates like a drumskin. That makes it, Batygin says, ideal for systems in which the central object is much more massive than the disk, such as protoplanetary disks and the rings of stars orbiting supermassive black holes. It will not work for galactic disks, however, like the spiral that forms our Milky Way.
But Ken Rice of The Royal Observatory in Scotland, who was not involved with the work says that in the scenario in which the central object is much more massive than the disk, the dominant gravitational influence is the central object. “It’s then not entirely clear how including the disk self-gravity would influence the evolution” he says. “My simple guess would be that it wouldn’t have much influence, but I might be wrong.” Which suggests the chief application of Batygin’s formalism may not be to model a wide range of systems but rather to make models for a narrow range of systems far less computationally expensive than N-body simulations.
Astrophysicist Scott Tremaine of the Institute for Advanced Study in Princeton, N.J., also not part of the study, agrees these equations might be easier to solve than those that describe the self-gravitating rings more precisely. But he says this simplification comes at the cost of neglecting the long reach of gravitational forces, because in the Schrödinger version only interactions between adjacent “wire” rings are taken into account. “It’s a rather drastic simplification of the system that only works for certain cases”, he says, “and won’t provide new insights into these disks for experts.” But he thinks the approach could have useful pedagogical value, not least in showing that the Schrödinger equation “isn’t some magic result just for quantum mechanics, but describes a variety of physical systems.”
But Saint Andrews’s Forgan thinks Batygin’s approach could be particularly useful for modeling black hole accretion disks that are warped by companion stars. “There are a lot of interesting results about binary supermassive black holes with ‘torn’ disks that this may be applicable to,” he says.