© Matt Strassler [April 28, 2014]
Let me start by asking you a couple of questions with simple, intuitive answers.
Let’s take a bowl and a marble (Figure 1). If
- I want the marble to remain stationary after I place it in the bowl, and
- I want to make sure it will remain in the same vicinity if I move the bowl slightly,
where should I put it?
Of course, I need to put it in the center, right at the bottom. Why? Intuitively, if I put it anywhere else, it will roll down toward the bottom, and swing back and forth. Eventually, friction will reduce the height at which it swings, and will bring it to a stop — at the bottom.
In principle, I could try to balance the marble on the edge of the bowl. But if I jiggle the bowl at all, the marble will lose its balance and fall off. So that’s not a place that satisfies the second criterion in my question.
Let’s call a position at which the marble can remain stationary, and from which it will not dramatically depart if the bowl or the marble are moved slightly, a “stable location for the marble.” The bottom of the bowl is such a stable location.
Ok, here’s another question. If I have two bowls, as shown in Figure 2, where are the stable locations for the marble? That’s easy too: there are two places, namely, the bottom of each bowl.
Finally, another one with an intuitive answer: If I place the marble in the bottom of bowl 1, and I then leave the room, seal it shut, ensure that no one goes in or out, and check that there have been no earthquakes or other disturbances, then what are the chances that, ten years later, when I open up the room again, that I would find the marble in the bottom of bowl 2? Of course, it’s zero. For the marble to move from the bottom of bowl 1 to the bottom of bowl 2, someone or something would have to pick up the marble and move it from one place to the other, carrying it over the lip of bowl 1, across to bowl 2, and over the lip of bowl 2. So it’s obvious the marble will still be in the bottom of bowl 1.
Obvious and essentially correct. Yet in the quantum world we live in, no object is ever quite stationary for more than an instant, nor is its location exactly knowable. So one of these answers isn’t quite true.
If (Figure 3) I put an elementary particle like an electron in a magnetic trap that acts like a bowl, tending to push the electron toward the center just the way gravity and the walls of the bowl push the marble toward the bowl’s center in Figure 1, then what is a stable location for the electron? Just as you would intuitively expect, the electron’s average position will be stationary only if I place the electron the center of the trap.
But quantum mechanics adds a wrinkle. The electron cannot remain stationary; there is a sense in which its location is subject to a sort of “quantum jitter”. This causes its position and its motion to be constantly changing, or (better) even to be undefined, by small amounts. [This is the famous ``uncertainty principle'' in action.] Only the average position of the electron is at the center of the trap; if you look for the electron, you’ll typically find it somewhere else in the trap, near but not at the center. And the electron is only stationary in the following sense: it’s typically moving, but its motion is in a random direction, and since it’s trapped by the walls of the trap, on average it goes nowhere.
That’s a bit weird, but it just reflects the fact that electrons aren’t what you think they are, and don’t behave like any object you’ve ever seen.
By the way, it also assures that the electron cannot be balanced on the edge of the trap, in contrast to a marble on the edge of a bowl (as in Figure 1, bottom). The electron’s position isn’t sharply defined, so it can’t be precisely balanced; and so, even without the trap being jiggled, the electron would become unbalanced and almost immediately would fall off.
But the weirder thing is what happens if I have two traps, separated from one another, and I put the electron in one trap. Yes, the center of either trap is a good stable location for the electron. That’s still true… in the sense that the electron can stay there and won’t run away if you jiggle the trap.
However, if I put the electron in trap number 1, and walk away, sealing the room etc., there’s a certain probability (Figure 4) that when I come back the electron will be in trap number 2.
How did it do that? If you imagine that electrons are like marbles, you will not be able to understand this. But electrons are not like marbles [or at least not like your intuitive notion of a marble], and their quantum jitter offers them an extremely small but non-zero probability of “walking through walls” — of going someplace that it would seem impossible for them to go — and ending up on the other side. This is called, poetically, “tunneling” — but you should not imagine that the electron digs a hole through the wall. And you’ll never catch the electron in the wall — in the act, so to speak. It’s just that the wall isn’t completely impermeable to things like electrons; electrons are not things that can be easily trapped.
Actually, it’s even crazier than this: because what is true for the electron actually is true for the marble in the bowl. The marble could end up in bowl 2, if you could give it enough time. But the probability of this happening is extremely extremely extremely small… so small that if you waited billions of years, or even billions of billions of billions of years, that still wouldn’t be enough. For all practical purposes, it will “never” happen.
The point is that our world is a quantum world, and all objects are made from elementary particles and are subject to the rules of quantum physics. Quantum jitter is ever-present. But for most objects that have a lot of mass compared to an elementary particle — a marble, for instance, or even a typical speck of dust — this quantum jitter is too small to observe, except in very specially designed experiments. And the consequent ability to tunnel through walls is also, therefore, never seen in ordinary daily life.
To say it another way: any object can tunnel through a “wall”, but the probability for it to do so typically goes down very rapidly if
- the object has a large mass
- the wall is thick (i.e. there is a long distance between its two sides)
- the wall is hard to penetrate (i.e. to punch through the wall in the usual way would require a lot of energy.)
For a marble to penetrate the lip of a bowl is possible in principle, but in practice might as well be impossible. For an electron to escape from one trap to another may be easy, if the traps are close together and the traps are not very deep, but it may be very difficult, if the traps are far apart or the traps are very deep.
Are We Sure Tunneling Really Happens?
Is this tunneling effect just speculation? Absolutely not. It’s fundamental in chemistry, takes place in many materials, plays a role in biology, and it is the principle used in our most clever and powerful microscopes.
For brevity, let me just focus on the microscope. In Figure 5 is a picture of atoms made using a scanning tunneling microscope, or STM. Such a microscope has a narrow needle whose tip moves just above the material that is being studied. (See Figure 6.) Both the material and the needle are, of course, made from atoms; and on the outskirts of the atoms are electrons. Roughly speaking, the electrons are trapped within the material under study, or trapped within the tip of the microscope. But the closer the tip is to the surface, the more likely the electrons are to tunnel from one to the other. A simple arrangement [for those who want to know: a voltage difference is maintained between the material and the tip -- and see further discussion in the next section] can assure that the electrons preferentially flow from the surface to the tip, and this flow is an electric current, one that can be measured. As the tip moves over the surface, and the surface is closer to or farther from the tip, the current changes, becoming stronger when the distance is smaller and weaker when the distance grows. By monitoring the current (or, alternatively, by moving the tip up and down to keep the current constant) as the tip sweeps across the surface, the microscope infers the shape of the surface, often with enough detail to see its individual atoms.
Tunneling plays many other roles in nature and in modern technology, which perhaps I’ll return to later. Soon I’ll discuss some of its implications in particle physics.
Tunneling Between Traps of Different Depths
In Figure 4, I implicitly assumed that the two traps were of equal depth — just as the bowls of Figure 2 were of the same shape. That means that an electron in either trap is equally likely to tunnel to the other one.
But let’s now suppose that one of the two traps for the electron in Figure 4 is deeper than the other — similar to imagining that one bowl in Figure 2 is deeper than the other. (See Figure 7.) Well, although the electron can tunnel in either direction, it is much easier for the electron to tunnel from the shallower trap to the deeper one than the other way round. Consequently, if we wait long enough, so that the electron has plenty of time to tunnel in either direction and back again, and then we start doing measurements to find out where the electron is, we will find the electron most often in the deeper trap. [There are some subtleties to this statement; the full story depends also on the shape of the trap, not just its depth, but this is good enough for now.] In fact, the difference in depths doesn’t have to be that large before the tunneling from the deeper to the shallower trap becomes extremely rare.
In short, tunneling will generally occur in both directions, but it is much more likely to occur from a shallow trap to a deep one (Figure 7.)
In fact, this feature is just what is used in the Scanning Tunneling Microscope to make sure the electrons tend to flow only in one direction, as indicated in Figure 6. Essentially, what is done is to make the tip of the microscope a deeper trap than the surface under study, so that the electrons prefer to tunnel out of the surface into the tip, rather than the other way round. [The microscope also works if the reverse is done, with electrons tunneling into the surface more often than out of it.] The traps are made deeper or shallower by using a battery, which creates a voltage difference between the tip and the surface, resulting in an additional energy difference between the electrons in the tip and the electrons in the surface. The fact that it is so easy to force the electrons to tunnel one direction more often than the other is part of what makes tunneling so practically useful in electronics.