Position and Momentum of Wavefunctions

In quantum mechanics, things are not described as particles. Nor are they described exactly like waves. They are described as wavefunctions. Wavefunctions look like this:

Well, they don't always look like that. It can look like almost anything, really. Also, a true wavefunction would occupy three dimensions, while this one only occupies one. But this is a nice clean example, in that we have a fairly good idea of where the object "is" and where it's "going".

Where is it? Well, the object doesn't really have a "location", strictly speaking. And yet we can measure the location anyway. We'll get an exact location, limited only by the accuracy of our measuring device. But that exact location may be here, or it may be there. It may be millions of miles away. Of course, some locations are more likely than others. The chance of it being millions of miles away is much smaller than the chance that Quantum Mechanics is simply wrong. It's most likely to be near the middle. The probability of finding it at any point is proportional to the amplitude-squared of the wave. The bigger the wave is, the more likely you are to find the object there.

The "expected" position is right in the middle. But our measurement will never be exactly in the middle. You'd have to be impossibly lucky for that to happen. It will always appear slightly off to the side. The average distance* from the middle is called Δx.

Where is it going? The object doesn't really have a "velocity", strictly speaking. And yet we can measure the momentum (equal to mass times velocity). We'll get an exact momentum, again limited only by the accuracy of our measuring device. How do we tell the momentum just by looking at the wavefunction anyway? It turns out that the momentum is related to how much the wavefunction goes up and down. The faster it goes up and down, the higher the momentum.**

Now, this is a little more difficult to visually realize, but the momentum that we measure is not exact. Just like when we measured position, there will be an "expected" momentum, but the measured momentum will always be slightly different from the expected. The average difference between our measurement and the expected momentum is called Δp.

The Leaky Faucet

It's time for an (imperfect) analogy. Let's say you're listening to the sound of a leaky faucet. It keeps dripping. I ask you how quickly it is dripping. Since you've been listening for a long time, you can give me a very accurate answer. But then I ask you how quickly it is dripping this minute. It might have been dripping faster or slower than average this minute. Well, you count the number of drips during the minute, and you tell me. But you might be off by a fraction of a drip. Therefore, your uncertainty would be 1 drip per minute. When I ask you how quickly it is dripping this second you will have an uncertainty of 1 drip per second. If I ask you how quickly it is dripping this instant, the question is nonsensical. There isn't any answer to give.

The wavefunction is analogous, with Δx relating the amount of time to listen to drips, and Δp relating to the rate of drips. The smaller Δx is, the greater Δp is, because you have less space over which to count the number of "drips". The smaller Δp is, the greater Δx must be, because you have to have lots of space to count the number of "drips" accurately. In general, we have the Uncertainty Principle:

Δx Δp ≥ ħ/2

This means that the product of the uncertainty in position and uncertainty in momentum are at least ħ/2, which is a universal constant of nature (equal to 5.27 x 10

^{-35}kg m^{2}/s--it is extremely small!). And yes, I can prove this mathematically from the axioms of Quantum Mechanics, though it is too difficult to do so here.You might ask, "Can't you just look at the faucet to see how quickly the water is accumulating into a drip?" Yes, but this is where the analogy breaks down. You can't do the same with a wavefunction. Well, yes, you can do a variety of experiments to determine exactly what the wavefunction is. And then, using this wavefunction, you can tell exactly what the expected position and expected momentum is. But if you directly try to measure either one, you won't get exactly the expected values.

Quantum Measurements

Recall that I said when you take a measurement, you get an exact value, only limited by the accuracy of your measuring device. Another important element is that once you've measured it, that measured value becomes the true value. For example, if you measure the momentum exactly, the wavefunction "collapses" into something like this:

In the wavefunction above (assuming it continues on to infinity), the momentum is exact. If you measure it again, the momentum will always be exactly equal to the expected momentum. However, the wavefunction above is also impossible. This wavefunction would be distributed evenly everywhere in the universe. Unless your device is capable of making the object be everywhere at once, it cannot measure the momentum exactly.

If you measure position exactly, you'll get this wavefunction:

In this wavefunction, the position is exact, at least for an instant. But the momentum is completely unknown. Asking for the momentum is like asking how quickly the faucet is dripping in a particular instant. The position won't stay exact for long because the wavefunction will quickly scatter in all directions. This wavefunction is also impossible because it has an infinite amount of energy. Unless you've got that kind of energy, you can't measure position exactly.

Although you can't measure position or momentum exactly, you can still measure them to any degree of accuracy you like, provided you have the instrument for it. What's to stop you from violating the Uncertainty Principle? Well, once you've measure it, you changed the wavefunction. If you measure it accurately enough, you drastically change wavefunction. First you can measure the position extremely accurately, and then the momentum extremely accurately. But if you try to measure position again, you may get an entirely different value than the first time.

This will occur no matter how good your measuring devices are. It has little to do with our common conception of uncertainty as the result of mere human error. It is a different, more fundamental kind of uncertainty that only really occurs on tiny, tiny scales. And that, my friends, is the Uncertainty Principle.

*technically, it's the root-mean-square distance

**The attentive reader might ask, "How do we know whether it's going left or right?" You can't, at least not from what I've shown. The wavefunction actually has two components, the "real" part and the "imaginary" part, and you need to see both to know which way the object is going.

## 0 comments:

Post a Comment