What is the Heisenberg Uncertainty Principle?
It's not really about uncertainty or measurement. It's just math.
I have an interest in interpretations of quantum mechanics. Lots of people know that quantum mechanics tells us something weird about reality, but it’s hard to articulate what weird thing it tells us about reality. And if you dig into this a bit, you’ll find yourself tangled up in a mess of difficult metaphysical questions that physicists have been sniping at each other over for a century or so. Quantum mechanics is good. It’s a theory that works. But past that, it’s really hard to say anything with any degree of certainty.
The most important thing to know about quantum mechanics is that it’s just math. It’s math that works, in the lab and in technology, but the essence of the theory is math. Quantum mechanics has its origins in an attempt to figure out how to explain some puzzling data about light and how light relates to the atomic structure of atoms that emit light. This is not about coming up with a picture or anything like that. We might see pictures in textbooks, but those are always the product of interpretation. What physicists do is try to come up with mathematical models that succeed in predicting and explaining observable data.
The basic principles of quantum mechanics were developed by Heisenberg and Schrodinger in the early 20th century. Heisenberg was the first to develop a mathematical model that fit the empirical data, but Schrodinger came out with another model shortly thereafter. These two models were eventually shown to be equivalent, but they were written in very different mathematical forms. Heisenberg was working with pure, uncut linear algebra. Schrodinger modeled his equations on the equations that were well-known as good mathematical models of wave motion. Of the two, Schrodinger’s equations were much more computationally tractable (and thus easy to work with), and also familiar from other physical work on waves. So while Heisenberg received credit for his advance, everyone quickly went to work with Schrodinger’s formalisms rather than Heisenberg’s. (Heisenberg was annoyed.)
One interesting thing about the mathematical models for describing waves is that they describe an inverse mathematical relation between ranges of values. For instance, waves diffuse when they pass through an aperture. That is, if a wave (of any kind, light, sound, ocean, whatever) passes through a narrow opening, the path of the wave won’t pass straight through the opening; it’ll spread out. That last sentence raises some obvious questions: How narrow an opening are we talking about here? And how much will the wave spread out? And the answer is that those two questions are related. The narrower the opening (relative to the wavelength of the wave), the greater the angle of diffusion. So if there’s a wide opening, there will be little to no diffraction. As the opening shrinks, the angle of diffraction will get larger and larger.
Schrodinger modeled his math for quantum mechanics on the well-known wave mechanics. So Heisenberg, working with the Schrodinger wave function, wondered: are there any of these inverse mathematical relations between ranges of values in the Schrodinger equation? There should be, if he’s just copying the wave math. And, lo and behold, there were these inverse mathematical relations. But interestingly, they hold between values that we typically don’t think of as values that have a range. The most well-known of these inverse relations is the relationship between the range of values for position and momentum. The wave function doesn’t give you a precise value for either the position or momentum of a particle. It gives a range of values for both, and the size of those ranges is inversely related. Narrow the range of the values for position, and range of values for momentum expands. Narrow the range of the values for momentum, and range of values for position expands. Mathematically, this is exactly the same thing as the value of the angle of diffraction expanding and contracting as the size of the diffusion aperture contracts and expands.
Now I’m sure you have plenty of questions about that. What does that mean, exactly, to say that the values of position and momentum have an inverse mathematical relation? And the answer is that there is no answer, or at least no uncontroversial answer. The uncertainty principle just is what I said. The mathematical value of the range of positions for a particle is inversely proportional to the mathematical value of the range of momenta for that particle. Period. Anything else, any further elaboration, is interpreting the uncertainty principle.
Indeed, even the name “uncertainty principle” is a bit of propaganda that is intended to make you think of a certain interpretation. For (we might think), if we nail down the position of a particle conclusively, via experiment, then the narrowing of the range of possible values of the position will shrink dramatically, and so the range of possible values of the momentum will increase dramatically. This renders the two values mutually unknowable. There is an ineliminable element of uncertainty in quantum mechanics.
But that’s not what the uncertainty principle says! These conclusions add on certain concepts that are nowhere in the math. “Measurement.” “Knowable.” “Uncertainty.” Those are all bits of interpretation. They are at the heart of what Heisenberg would later call the “Copenhagen interpretation” of quantum mechanics. But the Copenhagen interpretation is just that: an interpretation. Specifically, it’s a way of understanding the uncertainty principle in irreducibly epistemic terms. We might have called the “uncertainty principle” the “inverse ranges principle” or something like that. Calling it a principle about uncertainty, an epistemic notion? That’s your propaganda.
The principle itself is the math, which is a theorem provable from the Schrodinger equation. The Schrodinger wave function works, so something about it is right; it describes reality in some sense. What about it is right? In what sense does it describe reality? Those are controversial metaphysical questions. No one really knows.
So we should accept the wave function (provisionally) because it works. And so we should accept the “uncertainty principle,” because that’s just part of the math of the wave function. But that just means that we should accept a bit of math. If you’re trying to say what that math means about the world, you’re off the edge of the map.