It’s not usual in science for a known fundamental problem to remain at the heart of a discipline for a century. Usually these attract the focused attentions of lots of scholars and get solved. But quantum mechanics has resisted this — the combined attentions of some of the smartest minds ever — for all of this time. At its root, the problem seems simple enough: get the math, the ideal language to knit together phenomena in a logical way, to agree with experiment, such as the observed energy states of an electron bound in an atom, and get this to make intuitive sense. The last bit is the hard part, because here “intuitive” isn’t used glibly; specifically, it means figuring out why some of the predicted energy states are observed and others are not.
This will make more sense at the macroscopic scale of human experience. A good analogy might be the mathematical description of an apple falling off a ledge. Newton’s equations of motion give a good description of the position, speed, acceleration — more formally, momentum, its rate of change in time, and energy — of the apple, with no expectation that the process will spontaneously reverse and put the apple back up on the ledge. Countless experiments will confirm that the apple will fall from the ledge to the floor, not the other way around. More crucially, it doesn’t matter when you look at the apple to measure its momentum and energy — your observation won’t interfere with how it behaves. But as the spatial scale of the system is reduced to the atomic scale and below, things change. The electron transitioning between energy states may be thought of similarly to an apple falling from a ledge to the floor; but electrons behave in a fundamentally different way. First, any attempt to describe the electron as you have the apple will fail (you will find that its momentum and energy are quantized, so if you could “see” it over many trials, it would appear to jump rather than fall smoothly); and second, its position would depend on when and where you made your measurement. The reason for the first part is that electrons can only have certain amounts of energy, and energy comes in parcels (photons, in this case) — this is fine. It’s all very well understood and integrated into the math. The description of Erwin Schrödinger. he first part is that electrons can only have certain amounts of energy, and energy comes in parcels (photons, in this case) — this is fine. It’s all very well understood and integrated into the math. In the description of Erwin Schrödinger, who was the first to do this, the electron can be described as a sum (superposition) of waves of different frequencies, just like a sound, but with their frequencies limited to certain values (quantum states). When you plug values into Schrödinger’s Equation, the math works perfectly. (For hydrogen and helium, the simplest molecules, but presumably also for heavier elements if we had the computing power to carry out the computation. We don’t.)
But Schrödinger’s description is probabilistic. It gives the likely energy states of a (simple) quantum system, not the definitive final states that are measured in an experiment. Making sense of this connection between the probabilistic theoretical and definitive real worlds of physics is the crux of the problem. But it is more than philosophical. The result of the prediction is a measured quantity, a process that transforms a fuzzy statistical theory of possibilities into a measured state, as certain as an apple resting on a shelf or split on the floor. Formally, this is known as “collapsing of the wave function”. It follows from the so-called “Copenhagen Interpretation” of quantum mechanics, which emerged by about 1927 from Niels Bohr and Werner Heisenberg, and is the one I was taught. (In fact, there are many interpretations of quantum mechanics. I am only familiar with a few myself.)
All of that was to give some background to the article I’ve shared above, published now in Physical Review Letters (which is the gold standard for physics — sorry Nature!), not that I read it there.