The So-Called Heisenberg Uncertainty Principle

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Previously in series: Decoherence

As touched upon earlier, Heisenberg's "Uncertainty Principle" is horribly misnamed.

Amplitude distributions in configuration space evolve over time. When you specify an amplitude distribution over joint positions, you are also necessarily specifying how the distribution will evolve. If there are blobs of position, you know where the blobs are going.

In classical physics, where a particle is, is a separate fact from how fast it is going. In quantum physics this is not true. If you perfectly know the amplitude distribution on position, you necessarily know the evolution of any blobs of position over time.

So there is a theorem which should have been called the Heisenberg Certainty Principle, or the Heisenberg Necessary Determination Principle; but what does this theorem actually say?

At left is an image I previously used to illustrate a possible amplitude distribution over positions of a 1-dimensional particle.

Suppose that, instead, the amplitude distribution is actually a perfect helix. (I.e., the amplitude at each point has a constant modulus, but the complex phase changes linearly with the position.) And neglect the effect of potential energy on the system evolution; i.e., this is a particle out in intergalactic space, so it's not near any gravity wells or charged particles.

If you started with an amplitude distribution that looked like a perfect spiral helix, the laws of quantum evolution would make the helix seem to rotate / move forward at a constant rate. Like a corkscrew turning at a constant rate.

This is what a physicist views as a single particular momentum.

And you'll note that a "single particular momentum" corresponds to an amplitude distribution that is fully spread out—there's no bulges in any particular position.

Let me emphasize that I have not just described a real situation you could find a particle in.

The physicist's notion of "a single particular momentum" is a mathematical tool for analyzing quantum amplitude distributions.

The evolution of the amplitude distribution involves things like taking the second derivative in space and multiplying by i to get (one component of) the first derivative in time. Which turns out to give rise to a wave mechanics—blobs that can propagate themselves across space, over time.

One of the basic tools in wave mechanics is taking apart complicated waves into a sum of simpler waves.

If you've got a wave that bulges in particular places, and thus changes in pitch and diameter, then you can take apart that ugly wave into a sum of prettier waves.

A sum of simpler waves whose individual behavior is easy to calculate; and then you just add those behaviors back together again.

A sum of nice neat waves, like, say, those perfect spiral helices corresponding to precise momenta.

A physicist can, for mathematical convenience, decompose a position distribution into an integral over (infinitely many) helices of different pitches, phases, and diameters.

Which integral looks like assigning a complex number to each possible pitch of the helix. And each pitch of the helix corresponds to a different momentum. So you get a complex distribution over momentum-space.

It happens to be a fact that, when the position distribution is more concentrated—when the position distribution bulges more sharply—the integral over momentum-helices gets more widely distributed.

Which has the physical consequence, that anything which is very sharply in one place, tends to soon spread itself out. Narrow bulges don't last.

Alternatively, you might find it convenient to think, "Hm, a narrow bulge has sharp changes in its second derivative, and I know the evolution of the amplitude distribution depends on the second derivative, so I can sorta imagine how a narrow bulge might tend to propagate off in all directions."

Technically speaking, the distribution over momenta is the Fourier transform of the distribution over positions. And it so happens that, to go back from momenta to positions, you just do another Fourier transform. So there's a precisely symmetrical argument which says that anything moving at a very definite speed, has to occupy a very spread-out place. Which goes back to what was shown before, about a perfect helix having a "definite momentum" (corkscrewing at a constant speed) but being equally distributed over all positions.

That's Heisenberg's Necessary Relation Between Position Distribution And Position Evolution Which Prevents The Position Distribution And The Momentum Viewpoint From Both Being Sharply Concentrated At The Same Time Principle in a nutshell.

So now let's talk about some of the assumptions, issues, and common misinterpretations of Heisenberg's Misnamed Principle.

The effect of observation on the observed

Here's what actually happens when you "observe a particle's position":

Decoherence, as discussed yesterday, can take apart a formerly coherent amplitude distribution into noninteracting blobs.

Let's say you have a particle X with a fairly definite position and fairly definite momentum, the starting stage shown at left above. And then X comes into the neighborhood of another particle S, or set of particles S, where S is highly sensitive to X's exact location—in particular, whether X's position is on the left or right of the black line in the middle. For example, S might be poised at the top of a knife-edge, and X could tip it off to the left or to the right.

The result is to decohere X's position distribution into two noninteracting blobs, an X-to-the-left blob and an X-to-the-right blob. Well, now the position distribution within each blob, has become sharper. (Remember: Decoherence is a process of increasing quantum entanglement that masquerades as increasing quantum independence.)

So the Fourier transform of the more definite position distribution within each blob, corresponds to a more spread-out distribution over momentum-helices.

Running the particle X past a sensitive system S, has decohered X's position distribution into two noninteracting blobs. Over time, each blob spreads itself out again, by Heisenberg's Sharper Bulges Have Broader Fourier Transforms Principle.

All this gives rise to very real, very observable effects.

In the system shown at right, there is a light source, a screen blocking the light source, and a single slit in the screen.

Ordinarily, light seems to go in straight lines (for less straightforward reasons). But in this case, the screen blocking the light source decoheres the photon's amplitude. Most of the Feynman paths hit the screen.

The paths that don't hit the screen, are concentrated into a very narrow range. All positions except a very narrow range have decohered away from the blob of possibilities for "the photon goes through the slit", so, within this blob, the position-amplitude is concentrated very narrowly, and the spread of momenta is vey large.

Way up at the level of human experimenters, we see that when photons strike the second screen, they strike over a broad range—they don't just travel in a straight line from the light source.

Wikipedia, and at least some physics textbooks, claim that it is misleading to ascribe Heisenberg effects to an "observer effect", that is, perturbing interactions between the measuring apparatus and the measured system:

"Sometimes it is a failure to measure the particle that produces the disturbance. For example, if a perfect photographic film contains a small hole, and an incident photon is not observed, then its momentum becomes uncertain by a large amount. By not observing the photon, we discover that it went through the hole."

However, the most technical treatment I've actually read was by Feynman, and Feynman seemed to be saying that, whenever measuring the position of a particle increases the spread of its momentum, the measuring apparatus must be delivering enough of a "kick" to the particle to account for the change.

In other words, Feynman seemed to assert that the decoherence perspective actually was dual to the observer-effect perspective—that an interaction which produced decoherence would always be able to physically account for any resulting perturbation of the particle.

Not grokking the math, I'm inclined to believe the Feynman version. It sounds pretty, and physics has a known tendency to be pretty.

The alleged effect of conscious knowledge on particles

One thing that the Heisenberg Student Confusion Principle DEFINITELY ABSOLUTELY POSITIVELY DOES NOT SAY is that KNOWING ABOUT THE PARTICLE or CONSCIOUSLY SEEING IT will MYSTERIOUSLY MAKE IT BEHAVE DIFFERENTLY because THE UNIVERSE CARES WHAT YOU THINK.

Decoherence works exactly the same way whether a system is decohered by a human brain or by a rock. Yes, physicists tend to construct very sensitive instruments that slice apart amplitude distributions into tiny little pieces, whereas a rock isn't that sensitive. That's why your camera uses photographic film instead of mossy leaves, and why replacing your eyeballs with grapes will not improve your vision. But any sufficiently sensitive physical system will produce decoherence, where "sensitive" means "developing to widely different final states depending on the interaction", where "widely different" means "the blobs of amplitude don't interact".

Does this description say anything about beliefs? No, just amplitude distributions. When you jump up to a higher level and talk about cognition, you realize that forming accurate beliefs requires sensors. But the decohering power of sensitive interactions can be analyzed on a purely physical level.

There is a legitimate "observer effect", and it is this: Brains that see, and pebbles that are seen, are part of a unified physics; they are both built out of atoms. To gain new empirical knowledge about a thingy, the particles in you have to interact with the particles in the thingy. It so happens that, in our universe, the laws of physics are pretty symmetrical about how particle interactions work—conservation of momentum and so on: if you pull on something, it pulls on you.

So you can't, in fact, observe a rock without affecting it, because to observe something is to depend on it—to let it affect you, and shape your beliefs. And, in our universe's laws of physics, any interaction in which the rock affects your brain, tends to have consequences for the rock as well.

Even if you're looking at light that left a distant star 500 years ago, then 500 years ago, emitting the light affected the star.

That's how the observer effect works. It works because everything is particles, and all the particles obey the same unified mathematically simple physics.

It does not mean the physical interactions we happen to call "observations" have a basic, fundamental, privileged effect on reality.

To suppose that physics contains a basic account of "observation" is like supposing that physics contains a basic account of being Republican. It projects a complex, intricate, high-order biological cognition onto fundamental physics. It sounds like a simple theory to humans, but it's not simple.

Linearity

One of the foundational assumptions physicists used to figure out quantum theory, is that time evolution is linear. If you've got an amplitude distribution X1 that evolves into X2, and an amplitude distribution Y1 that evolves into Y2, then the amplitude distribution (X1 + Y1) should evolve into (X2 + Y2).

(To "add two distributions" means that we just add the complex amplitudes at every point. Very simple.)

Physicists assume you can take apart an amplitude distribution into a sum of nicely behaved individual waves, add up the time evolution of those individual waves, and get back the actual correct future of the total amplitude distribution.

Linearity is why we can take apart a bulging blob of position-amplitude into perfect momentum-helices, without the whole model degenerating into complete nonsense.

The linear evolution of amplitude distributions is a theorem in the Standard Model of physics. But physicists didn't just stumble over the linearity principle; it was used to invent the hypotheses, back when quantum physics was being figured out.

I talked earlier about taking the second derivative of position; well, taking the derivative of a differentiable distribution is a linear operator. F'(x) + G'(x) = (F + G)'(x). Likewise, integrating the sum of two integrable distributions gets you the sum of the integrals. So the amplitude distribution evolving in a way that depends on the second derivative—or the equivalent view in terms of integrating over Feynman paths—doesn't mess with linearity.

Any "non-linear system" you've ever heard of is linear on a quantum level. Only the high-level simplifications that we humans use to model systems are nonlinear. (In the same way, the lightspeed limit requires physics to be local, but if you're thinking about the Web on a very high level, it looks like any webpage can link to any other webpage, even if they're not neighbors.)

Given that quantum physics is strictly linear, you may wonder how the hell you can build any possible physical instrument that detects a ratio of squared moduli of amplitudes, since the squared modulus operator is not linear: the squared modulus of the sum is not the sum of the squared moduli of the parts.

This is a very good question.

We'll get to it shortly.

Meanwhile, physicists, in their daily mathematical practice, assume that quantum physics is linear. It's one of those important little assumptions, like CPT invariance.

Part of The Quantum Physics Sequence

Next post: "Which Basis Is More Fundamental?"

Previous post: "Decoherence"

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