Sometimes when scientists encounter complex and vexing problems that require mathematics, they are forced to occasionally bend the rules of math and follow strange steps and solutions that work, but which don’t fully make sense at the time.
One such case of weird workmanship in the world of quantum mechanics was recently brought to light by BYU physics professor Justin Peatross.
Peatross wrote an article that was recently published as the cover story in an issue of the American Journal of Physics (AJP). His article discusses Walter Gordon’s famous 1926 article on calculating photon scattering by a free electron and Gordon’s both incorrect and correct math.
While calculating the problem, Gordon altered the equation from solving for the square of an integral to solving for the integral of the squared integrand. This not only completely changed the problem but disregarded the rules of integration in getting there.
“He did this funny thing that had no mathematical justification, but he realized he needed to do that,” said Peatross. “It wasn’t until a decade later, with the invention of quantum electrodynamics, that his change was put on a proper foundation.”
Since quantum electrodynamics is computationally unwieldy, many scientists introduce assumptions and simplifications that are based on classical electrodynamics. This is particularly true in the high-intensity laser community.
One common simplification is the treatment of electrons as diffuse quantum waves when computing radiation. In fact, even though electrons exhibit wave-like behavior, they still radiate like point-charged particles. Peatross’s article shows how and why classical electrodynamics leads to inconsistent conclusions in certain circumstances and how Gordon surreptitiously sidestepped the issue.
“The gist of the article is that we do an analysis of classical electron waves and how an electron behaving that way would emit light. The takeaway is that it doesn’t act like quantum electrodynamics says it is going to act,” said Peatross.
As often as solutions to problems make sense, there are sometimes solutions that don’t make immediate sense. Additionally, history has demonstrated that it is necessary to problem-solve your way out—and even sometimes move the square for no reason.