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.