Saturday, November 14, 2009

Where the Uncertainty Principle Really Comes From

If you have read much popular physics literature, chances are you have encountered this formulation of the Heisenberg Uncertainty Principle:
"The measurement of position necessarily disturbs a particle's momentum, and vice versa". This "measurement" formulation comes from Heisenberg himself and has lodged itself firmly in the popular imagination; indeed, it has spread beyond physics to become an intellectual paradigm used in all sorts of contexts.

Unfortunately, the measurement formulation is a very inaccurate explanation of the Uncertainty Principle. It is not false, exactly, but it is circular and does not get to the root of the phenomenon. Ironically enough, it is probably more correct in its applications outside the field of physics than it is for the actual physical phenomenon.

For readers who have grown comfortable with the measurement formulation, I will now explain why it is inadequate; those who wish may skip straight to the next section where I present a more accurate explanation.

The Problem with the "Heisenberg Microscope"

The "measurement formulation" of the Uncertainty Principle was invented by Heisenberg himself, who illustrated it with a famous thought experiment known as the "Heisenberg Microscope".

The setup is simple: a scientist is trying to locate a particle by means of light. In other words, he or she is trying to figure out where the particle is by bouncing light off of it and observing the reflected light. This is just the normal operation of a microscope, but we imagine applying it at the tiny scale of individual particles.

What our scientist finds is that the particle's location is not easily pinned down. He or she may try to increase the precision of the observation, but the effort becomes self-defeating, and in the end the scientist must be satisfied with limited knowledge.

In more detail, the way to increase the precision is by using a shorter wavelength of light. Shorter wavelengths allow the resolution of smaller distances (the "Rayleigh Limit", which I will discuss in a subsequent blog post).

But now the quantum rears its enigmatic head, for light, as we know now, is made up of discrete particles known as photons, and photons of lower wavelength have higher momentum. (This was the 1905 discovery for which Albert Einstein won the Nobel Prize. If momentum is an unfamiliar concept, you can substitute "energy" instead for the purpose of this explanation).

So the shorter the wavelength we try to use, the higher the momentum of the photons - and the more they knock around the particle we are trying to observe. By using very short wavelengths, we can know very precisely where it was when the photon hit it - but only at the cost of losing any idea where it is afterwards. Conversely, if we use very long wavelength photons, we will get only a vague idea about where the particle was - but at least we will know that it didn't get knocked away from there.

And voila, the "Uncertainty Principle": we can't observe the particle without disrupting it. It sounds great - once one accepts all the things we said about photons.

Therein lies the rub, which makes this explanation circular. The problem arose because shorter wavelength photons have higher momentum. But why is this? Why are there no short wavelength, low-momentum photons which we could use to nail down our particle definitively?

The reason is that the Uncertainty Principle applies to photons too. Photons whose location can be known precisely - i.e., short-wavelength photons - necessarily have a big uncertainty in momentum, which is essentially the same as saying they have high momentum (because something with definitely low momentum cannot have much momentum uncertainty).

So the Heisenberg Microscope "explains" the Uncertainty Principle for other particles only by assuming the same principle for photons. It is a circular explanation.

The True Origin of the Principle

The true origin of the Uncertainty Principle lies at the heart of Quantum Mechanics (QM) itself. It is deeper and more interesting than the "measurement" explanation - but also a bit more abstract and mathematical. We will have to look at a couple of graphs to understand it.

The key to the Uncertainty Principle is that position and velocity (technically, momentum) are not separate in QM. In Classical Mechanics (CM) position and velocity are just two sets of numbers with no connection to each other at a given moment. Of course, the velocity tells how the position will change in the next moments - but that doesn't change the fact that particles can have any position and any velocity at one particular moment.

In QM the situation is very different. Neither position nor velocity is a fundamental quantity in QM; rather, every particle is defined by a "wave function", symbolized by Ψ. Ψ gives the probability that the particle might be seen at different positions. It can be visualized as a simple graph showing probability vs. position; two examples are shown in Figure 1 below:

Figure 1
Two examples of the particle wavefunction, Ψ.
A. Position more certain, velocity less certain
B. Position less certain, velocity more certain

(Actually I have simplified the situation slightly, because Ψ really is a complex-number function and its square is the probability. This is incredibly important for physics but not for understanding the Uncertainty Principle!)

The graph shown in Fig. 1A depicts a particle whose position is relatively certain. We can tell this because the graph is very narrow, meaning that the probability of seeing the particle is concentrated in a small region. Conversely, Fig. 1B is wide, and depicts a particle having very uncertain position.

Already here we can see that to have complete certainty about position is an unusual, even pathological case in QM. For complete certainty, the particle's graph would have to be so narrow that it covered only one point of space, being zero everywhere else. Such a weird graph isn't going to arise in a normal physical situation.

So the particle's position information is given by the graph of Ψ. Where is the velocity information? The sensible thing would be to have another wavefunction for velocity; however, nature doesn't always choose to be sensible!

No, it turns out that the velocity information is magically encoded right into the position wavefunction. This one graph gives us both position and velocity, making these two quantities indivisible in QM (the technical term is "complementary"). This close relationship is the root of the Uncertainty Principle.

The encoding of velocity information in Ψ is simple but not at all obvious. Velocity is represented, roughly speaking, by the steepness of the slopes on the graph. A graph with steeper slopes - a "bumpier" graph - encodes higher velocities than a smooth, non-bumpy graph. Of course it is not one particular velocity which the graph encodes, but rather the probabilities of different velocities, just as with position. Bumpier graphs have a higher spread of velocities than smooth graphs.

Now if look back at Fig. 1, we can see exactly where the Uncertainty Principle is coming from. Figure 1A, having the more certain position, also has much steeper slopes than Fig 1B. Therefore, the velocity of the particle in Fig 1A is less certain than that of Fig 1B.

This simple example shows the essential rule: the more we try to squeeze the particle's position - as in Fig. 1A - the steeper the slopes on its graph, and the more uncertainty is present in velocity. Conversely, the more we try to pinpoint the velocity - which means smoothing out the slopes in the graph, as in Fig. 1B - the wider the graph grows, and the less certain the position becomes.

This then is the true origin of the Uncertainty Principle. It is not related to the process of measurement, which should not be surprising given that measurement is also necessary in Classical Mechanics. Rather, the Principle comes from the very different mathematical definition of a particle in Quantum Mechanics, the basics of which were sketched above.

The fact that position and velocity are united in Quantum Mechanics gives rise to no end of surprising phenomena, and almost seems to suggest that the concepts of space and motion might be unified in the underlying theory of spacetime, whatever that may be. It seems a bit gratuitous to have a spacetime capable of supporting independent particle velocities and positions, when the particles themselves don't possess them.