Statistical picture of Quantum Mechanics | Locklin on science

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Statistical picture of Quantum Mechanics

Posted in physics by Scott Locklin on June 27, 2026

The statistical picture of quantum mechanics is an idea I had in my mind more or less as the correct picture, while occasionally simultaneously holding the gaseous nonsense that the Schroedinger equation somehow describes things the individual particles are doing. Stated simply, the modulus of the solutions to the Schroedinger equation describe a statistical ensemble of many similarly prepared quantum systems.  That’s what the Born rule says, and that’s what the experimenter always measures. The quantum weirdness nonsense mostly comes from the idea that the wave function has some independent reality of its own, and that it is describing the motion of an individual electron or hydrogen atom or whatever. Copenhagen (sort of) and all the other weird interpretations of quantum mechanics say this. They also say the wave function collapses somehow (or we split off in the multiverse, which is even more absurd), which is actually not something in quantum mechanics. That’s something pasted on afterwords.

Imagine you’re doing some classical quantum experiment, say, electron diffraction. You can detect a single electron on the multi-channel plate, but you’re only going to see the weird interference pattern when there are a whole bunch of electrons. Similarly when I do electric discharge on some hydrogen, I get the Balmer series in my spectroscope, the image of the Balmer series shows up on my photographic plate or CCD camera in a statistical way. There are no wavefunctions of individual electrons collapsing when it hits the detector as in Copenhagen or whatever Multiverse deranged acid trip view of things. The individual electrons or photons or whatever go about their business, and you only see weirdness when you look at a whole bunch of them. Some people say the individual particles have no particular individual states as they fly through the void, many others say they do. I’m inclined towards "they do" based on observation, but maybe that makes me a weirdo.

I’m on well trodden ground here: the literal and exact interpretation of the Born rule was the view of Einstein which he took great pains to state clearly, and which many of his colleagues (including Born and Bohr) actually agreed with, but they also disagreed with for weird reasons unknown to me. Perhaps he stiffed them on a dinner check or pinched their wives bottoms, or perhaps they just misunderstood each other. We know one of the Born-Einstein debates on this, Born was straw-manning something that was never said, adjudicated so by Pauli. It doesn’t really matter; most people’s views of the actual history of what uncle Albert believed is as false as their views of the Great Depression or World War-1, which were roughly contemporary events. Here’s what he clearly said:

It seems to be clear, therefore, that Born’s statistical interpretation of quantum theory is the only possible one. The Ѱ function does not in any way describe a state which could be that of a single system; it relates rather to many systems, to “an ensemble of systems” in the sense of statistical mechanics. If, except for certain special cases, the Ѱ function furnishes only statistical data concerning measurable magnitudes, the reason lies not only in the fact that the operation of measuring introduces unknown elements, which can be grasped only statistically, but because of the fact that the Ѱ function does not, in any sense, describe the state of one single system.

Then Einstein again:

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

Looks right to me, and I don’t see any experimental proof that the situation is otherwise. It’s possible there could be such an experiment demonstrating that wave functions apply to individual quantum particles somehow, but nobody I know of is thinking about doing such a thing (PBR is fake and gay and doesn’t count).

At some point everyone reading this who has actually looked at quantum mechanics to the point of solving differential things, you will have seen the Hamilton-Jacobi equations. If you take the Schroedinger equation to the classical limit  (ℏ→0), it is the Hamilton Jacobi equation. The Hamilton-Jacobi equations are also to be interpreted statistically. It’s solving for equal-action surfaces: that’s the wavefront piece. The ℏ piece represents our ignorance of the "microstates" of the individual particle: otherwise Schroedinger is giving us the equal-action surfaces like in the regular Hamilton-Jacobi picture. That’s why the solutions to energy conserving...