Newton's birthday (in the Julian calendar) is Sunday, so we're in the final days of the advent calendar. Which means it's time for the equations that are least like anything Newton did, such as today's:
This is the Schrödinger equation from non-relativistic quantum mechanics. If you want to determine the quantum state of an object that's moving relatively slowly, this is the equation you would use.
It also has probably the greatest origin story of any of the equations we've talked about. Or at least the most salacious origin story of any of the equations we've talked about...
Erwin Schrödinger (picture from Wikimedia) was a famous womanizer-- after he left his native Austria to get away from the Nazis, he lost a job at Oxford because he was openly shacking up with the wife of a colleague-- but he outdid himself when he was working on the equation that now bears his name. He had been struggling with the problem for a while, and eventually took off for a vacation with one of his many mistresses. Historians aren't sure of the exact identity of the woman, because while he kept detailed notes on all his assignations, the journal covering this period has gone missing. There's a Tim Powers novel in that, somewhere.
What is known is that while he was there, he spent his free moments thinking about physics-- one version of the story has him spending the days skiing, then staying up late at night doing physics-- and worked out the above equation. This introduced the quantum wavefunction to physics, and allowed him to correctly predict the energy states of hydrogen, giving a more rigorous basis for Bohr's model.
Werner Heisenberg had previously worked out a complete version of quantum mechanics using matrix mathematics, but this was not particularly popular with physicists, who at that time did not regularly use matrices. Schrödinger's equation is a differential equation similar to that describing classical waves, and as such was much more familiar, and quickly adopted by just about everybody. There was a bit of a rivalry between the two versions for a while, before they were shown to be equivalent. Nowadays, every physicist learns about matrices, so the two versions are somewhat interchangeable-- the version above uses Dirac's state-vector notation, which is in some ways a matrix version of the wavefunction-- and it's sort of hard to separate them.
So, what's interesting about this? Lots of things, starting with the i on the right-hand side, which represents the square root of negative one, in imaginary number. This means that solutions of the Schrödinger equation are necessarily complex, having both real and imaginary parts. This is responsible for a lot of the odd aspects of quantum physics-- because we only ever measure real values, we need some extra interpretive layers with quantum theory, which leads to the Born rule that the wavefunction squared gives the probability of finding a particle at a particular position.
This is the point where Einstein and, ironically, Schrödinger himself, parted company with quantum theory. Neither of them liked the probabilistic nature of the theory, and the famous cat thought experiment (Emmy's very favorite) was invented precisely to show the logical absurdity of this approach. Of course, these days, we know that all of the weird predictions are absolutely and unequivocally true, but it's taken a great deal of work to get to that point.
And what's this good for? Well, pretty much everything. The Schrödinger equation is the first really great calculational tool for quantum physics, and enabled physicists of the 1930's to start making accurate predictions about all sorts of systems where they previously had relied on an ungainly assortment of ad hoc rules to get approximate results. It was soon followed by Dirac's relativistic equation (Dirac, Schrödinger, and Heisenberg all got Nobel Prizes in 1933), which allowed even better precision, and started physics on the path to QED and the phenomenal success of quantum theory.
And quantum mechanics is behind just about every good thing in modern technology. You couldn't make transistors without an understanding of the quantum nature of the electron, so all of consumer electronics can be traced to this equation. You couldn't make lasers without understanding the quantum nature of matter, so the fiber-optic Internet can be traced to this equation. Everything good about modern physics starts here.
So, take a moment today to appreciate the most fruitful booty call in the history of science, and come back tomorrow for the penultimate equation in our countdown to Newton's birthday.
But more importantly, SchrÃ¶dinger discovered a method by which someone as dorky as Erwin SchrÃ¶dinger could get laid more often than the captain of the football team.
Unfortunately, he never documented the procedure, so the rest of us physics dorks are still completely clueless when it comes to getting girls...
jdl: Feynman eventually re-discovered and published that material.
The SchrÃ¶dinger equation is about more than measurable probability. It represents a certain amount of mass-energy, and momentum as well. Hence, it is false (at least, doubtful) that the SE can just "continue evolving" (as e.g. in MWI) despite detection events. That would mean, the entire amount of mass-energy of the original particle ends up in multiple locations, as the wave is "contracted" to that point in an observation. Sure, the cat result is "absurd" but that means that either measurement really is special and something causes real collapses, or that we really can't adequately represent things at the quantum level. Letting the SE continue to evolve as something "real" is entertained by some, but is doubtful as I outline below.
I don't agree that any excuse, loss of interference or whatever, can resolve the contradiction. It is still "more" all over in total sum. Also lack of coherence really should just mean a messier wavefunction that stays "wave" until collapsed, *then* produces a type of statistics that coincidentally resembles "classical statistics" but is not a stand-in for same. (Note that if we really could detect amplitudes directly, decoherence would just scramble distribution of amplitudes but not change the logical character of the wave as distributed "wave." And if we can't, then something "else" has to happen anyway. If anyone thinks that loss of coherence is what allows for exclusionary localizations, then why do we still find statistics to show quantum interference (as in double slit) as well?
Happy Solstice, folks.
Unfortunately, he never documented the procedure
There wasn't room in the margin.
Neil Bates: The one place where that really creates a problem is if you try to make quantum mechanics play nice with classical general relativity. Because we don't yet have a well-developed way of making spacetime itself "superposed" in the same way as all the quantum-mechanical matter and energy in the experiment, there's not a consistent way to couple one to the other.
I think most cosmologists would say it's a demonstration that quantum gravity is really necessary--in a quantum world you have to go all the way, you can't ultimately be semiclassical.
Some people have tried to go the opposite way, and argue that wavefunction collapse happens when the difference between possibilities crosses some threshold of gravitational significance (Roger Penrose was arguing for this in popular books back in the late Eighties). But I don't know how precise these ideas have been made.
Thanks Matt but details of space-time curvature etc. are not germane to the basic point, that a given quantity of mass-energy spread out (if you - or maybe I should say, Mother Nature Herself - are a realist) and then it's all accounted for "here" when we find the same quantity. Sure, various things about gravity etc. may account for it being able to localize, but the more basic issue is that it needs to localize - either all of it one place, or "thinned out" all over (again, to a realist - I prefer to just accept that our wanting to be able to represent everything there is, means we are able to), but not both as some misguided souls imagine.
BTW, is the Schroedinger Equation the first time imaginary numbers actually part of a physical equation per se? AFAIK, it is - that is cool.
"Fine minds make fine distinctions."
The mass-energy does not spread out, the probability does.
The word you are looking for is "reification".
Oddly, this issue is rarely raised with respect to light "waves", only with "matter", even when using a quantum device to make a photograph. Somehow we seem to have less trouble with photons because we don't imagine light as acting like baseballs.
And imaginary numbers were used for the impedance that is central to AC circuits. Although I don't know when that notation became common, it goes back to the basic way of solving second order differential equations (generating the characteristic equation).
Wouldn't it be the wavefunction times its complex conjugate, rather than squared?
CCPhysicist, the entire WF is not just the probability of the particle "being there" - it does show the momentum etc. as per the frequency and momentum relation, etc. If you say it's just "the probability" that spreads out, are you saying you believe in Bohmian mechanics: the particle "really is" in a definite place, definite trajectory, just not knowable and "guided" to simulate interference with itself? If not localized "in flight", if we strive for "realism", then the mass-energy is indeed spread out. That actually has relevance to things like electrons in atoms: the electron cloud effectively represents a fuzzy spread of charge and thus a "density-like_ source of field, or else the field would fluctuate and there'd be radiation. It's not just an abstraction about finding things. Of course, we don't have to be realists, we can say there's no authentically real way to represent things between interactions.
Furthermore, even if we think of "probability of being there", that still means "of being here and not somewhere else" - the whole program of MWI, to have it both ways, so to speak, violates the whole idea by having the effective concentration of the entire mass-energy in multiple locations.
Merry Christmas Eve!