I Used to Do Something That Was Almost Like A Real Job. . .

Image of Jeremy taken shortly after his thesis defence.

So, it finally happened. My paper, "Quantum tunneling and reflection of a molecule with a single bound state," has been published in Physical Review A. I promised in a previous post that I would write a summary of paper. So, with reading week quickly evaporating away, about eighty physics assignments yet to be marked, and a Japanese midterm on Monday, I thought I should get on that.

The approach I decided to take was a basic parsing/elaboration of the abstract, since that's the only part of the paper available to read on the internet without having to pay a fee. The abstract, by definition, pretty much lays out the content of the paper anyway, so with a bit of explanation you should be able to get the jist of what we, i.e. myself, Danielle Kerbrat, and my supervising prof Dr. Mark Shegelski, discovered and published. I'm going to assume that anyone who reads this has about high-school level science education, which means I'll have a lot of explaining to do.

Abstract:

In this article, we present the results of studies on the quantum mechanical tunneling and reflection of a diatomic, homonuclear molecule with a single bound state incident upon a potential barrier.

Hoo-boy. Where to start?

The "diatmoic, homonuclear molecule" is basically a pair of identical particles that interact with and, loosely speaking, "attach" to one another by means of an attractive force. Usually, the particles in question are atoms. However, our formulation is general enough to be applied to any pair of "attached" particles, such as Cooper pairs and excitons. These examples appear later in the abstract, so I'll explain what they mean later on.

When the atoms are attached to each other, we say that the molecule is in "bound state." When they aren't, we say they're in an "unbound state." To be in a "bound state," the atoms in the molecule must have lower total energy than two free atoms. To understand what that means, imagine a you're in a region that's entirely flat except for a small, bowl-shaped valley. If you're in the valley, you have to expend energy in order to get out of the valley. If you don't have enough energy to climb out of the valley, you're stuck-- "bound" to the valley. Another way of thinking about this is that, if you're standing in the valley, you have less energy than if you're standing in the flat plain. When two atoms are "attached" to each other in a molecule, what's really happening is that the force they exert on one-another creates a sort of potential energy "valley," whereas two free atoms are in a potential energy state more akin to standing in the flat region outside of the valley.

So what does it mean for a molecule to have a "single bound state?" In order to understand the behaviour of small objects, like atoms, molecules, electrons, etc. we had to discover a whole new set of physicals laws, which we call quantum mechanics. The problem with quantum mechanics is, well, it's weird. One of the implications of quantum mechanics is that, if two atoms are bound in a molecule, then they can only occupy certain energy levels-- we say that the energy levels are "quantized," hence "quantum mechanics." Think back to the valley for a minute. You could stand at the very bottom of the valley, or half-way up the valley, or two-thirds of the way up, or one-quarter, or any other place you like. With any given height up the valley, there is a corresponding potential energy level. In other words, the laws of physics do not restrict you to one or another given energy level in the valley. However, if the valley were like a molecule, you could only occupy certain specific energy levels. You could, say, be at the very bottom, or half way up, or two-thirds of the way up, but you could not be at any other altitude. When you're standing at one of the permitted altitudes, you could be said to be in one of the given "bound states" of the valley. Likewise, the atoms in the molecule can only exist in certain bound states. What these states are depends on the kind of molecules we're considering. For our paper, we consider a molecule whose parameters are such that there is only one bound state. If we go back to the example of the valley, that would mean that we can only stand at the very bottom of the valley-- no other altitude is permitted. One more thing that I may as well mention now is that the title of the paper mentions that we're considering a "weakly bound" molecule. This is akin to a very shallow valley. The implications of weak binding will be made more clear later on, so I'll leave it for now.

The other important thing mentioned in the above excerpt is the idea of "quantum tunneling." Purge your mind of the valley, for now I'm going to ask you to imagine you're riding a bike toward a hill. I'm also going to ask you to imagine, for the sake of argument, that once you start climbing the hill you stop pedalling you bike. If you were going fast enough before you started climbing, then you'll have enough kinetic energy to coast over the top of the hill and reach the other side. If not, you'll come to a stop before the crest of the hill and begin rolling back down. This makes sense, so of course quantum mechanics has to find some way screw it up. The way it does this is through the phenomenon of quantum tunneling (since my paper was published in an American journal, I will continue to spell it as "tunneling," and not "tunnelling").

What I'm about to tell you is strange, but since I'll have to discuss it eventually, and since it does have bearing on the explanation of quantum tunneling, I figure I may as well get it out of the way now. Do you remember in science class when you were taught that light behaves as a wave? Do you also remember hearing somewhere or reading somewhere that light is composed of particles called photons? Did you ever step back and wonder why scientists just can't seem to make up their bloody minds on the issue? Is light a wave or a series of particles? It must be one or the other, it can't be both. Well, according to quantum mechanics, light is both a wave and a series of particles. . . and so is everything else! Electrons, protons, atoms, molecules, your computer, you yourself. . . all waves. "But waves of what?," you might ask. Probability. Basically, the wave part of a given object, be it a photon, electron, atom, or molecule, determines the probability of observing that object at a given place (it also gives the probability of the object having a given momentum, but that's a whole other story). I'm oversimplifying a bit, but at any position where there's a crest in the wave, the probability of observing a particle at that position is high; wherever there's a trough, the probability is low.

This complicates the study of physics at the microscopic level quite a bit. Since the days of Newton, physics has always used particles to understand the laws of motion, with the implicit assumption that we can always take a measurement or make an observation and determine where the particle is at any given time. Additionally, if we know exactly where a particle is, what its speed and direction of motion is, and all of the forces acting on it are, it was assumed that the laws of physics could be used to predict its position and velocity at any time, past, present, or future. It was assumed, in other words, that the laws of physics act in a deterministic way. Quantum mechanics, however, says that, if we think in terms of particles, the laws of physics must probabilistic. But this means that we cannot use physical laws to make any solid predictions about the behaviour of a given object, rendering those physical laws next to useless. However, it turns out that if we think instead in terms the probability waves mentioned earlier, we have a lot more luck. Unlike particles, probability waves do behave deterministically. Understanding just how these waves behave allows physicists to make some very interesting, very counter-intuitive predictions.

In the macroscopic world that we all live in, this doesn't really amount to much. Even though there is a probability wave associated with each of us, the probability of any of us being exactly where we are is 100%. At the microscopic level, however, this becomes much more pronounced. One example of how much more pronounced it is quantum tunneling. Recall the proverbial hill I discussed earlier. The microscopic equivalent to the hill is something called a "potential barrier." Imagine some microscopic particle approaching a potential barrier with some given kinetic energy. If it behaved the same way as the bike climbing the hill, then the particle would definitely pass if it had high enough kinetic energy, and would definitely not pass if it didn't. But, you'll recall, nothing is "definite" as far as particles are concerned, and in order to make predictions we have to think in terms of the wave, or "wave function" in physicist parlance, associated with the particle. It turns out that, no matter what the energy of the incoming particle, a chunk of the wave will always manage to travel past the barrier. What this means is that, no matter what the energy of the incoming particle, there is some probability that the particle will be observed on the other side of the barrier. This is like the bicycle appearing on the other side of the hill even though it was only going fast enough to make it half way up-- the only way this could happen is if the bicycle travelled through a tunnel in the hill. Hence, "quantum tunneling." Make no mistake, though, the particle didn't "dig" its way through the potential barrier. Rather, the laws of quantum mechanics allowed the particle to travel through the barrier as though it were not there at all.

If we're only considering a single particle incident upon a given barrier, then it's relatively easy to calculate the wave function and thus find the probability of tunneling. However, when we start to consider more complex objects like, say, a diatomic homonuclear molecule, things get very ugly. Instead of one particle, we now have to consider two, which means we have to consider the object as having size and being spread out in space. Moreover, these two particle are being affected not only by the potential barrier but by the force attracting them to each other. This attractive force creates a "potential well"-- the microscopic equivalent to the metaphorical valley-- which must be taken into account as well. Recall also that the molecule can exist in any one of a number of bound or unbound states. As a result the molecule can undergo changes of state upon interacting with the potential barrier. These factors complicate things so much that the tunneling of molecules wasn't seriously investigated until 1994. Quantum tunneling of single particles, on the other hand, has been investigated since the 1920's.

From the next part of the abstract:

In the first study, we investigate the tunneling of a molecule using a time-dependent formulation. The molecular wave function is modeled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration.

(Our paper is actually a combination of two different studies. We had initially intended to publish two papers, but due to various circumstances we decided to publish both studies in a single paper.)

In quantum mechanics, you can look at things in either a "time-independent" way or a "time-dependent" way. For the purposes of describing the results in the paper, the difference between the two formulations is as outlined as follows.

In studies of quantum tunneling, we're usually interested in calculating the probability that a given object will be observed ahead of the barrier-- "probability of tunneling"-- or behind the barrier-- "probability of reflection". The time-independent formulation is very useful for calculating these probabilities, but it's not useful for describing what happens to the molecule as it's tunneling through the barrier. In order to study this, the so-called "tunneling dynamics," you need to use a "time-dependent" formulation. The problem is that this is quite a bit harder to do than using a time-independent formulation. For that reason, every study (that we're aware of) in molecular tunneling that came before this paper used a time-independent formulation. In other words, to my and my co-authors' knowledge, this paper is the first to use a time-dependent formulation to investigate the tunneling of molecules, making me and my co-authors the world's foremost experts in time-dependent molecular tunneling!

What's that, Alexandre Bilodeau? You're the first Canadian to win gold at the Winter Olympics on home soil? Big whoop.

Anyway. With a time-dependent formulation, we basically created a computer simulation of the molecule's wave function and calculated how the wave function behaves as it interacts with a potential barrier. That, in a nutshell, is what all that talk about "Gaussian wave packets" and "Crank-Nicholson integration" is referring to. It was a very difficult calculation. Like all previous work done at UNBC on molecular tunneling, we had to use the university's supercomputer in order to run the simulations. So what do we have to show for it?

We found that the molecule could take one of multiple paths once it begins to interact with the barrier. For one, it could reflect. Basically, the molecule hits the barrier, temporarily breaks apart (i.e. transitions to an unbound state), recombines, and bounces back from the barrier. This isn't really a surprising result. But a couple of the other paths it could take are surprising.

From the abstract:

It is found that a molecule may transition between the bound state and an unbound state numerous times during the process of reflection from or transmission past the barrier.

This means that, if the molecule follows a path such that it does tunnel through the barrier, it will break apart and recombine some number of times before it passes the barrier. The reason we think this happens is summarized, in highly simplified fashion, as follows. We chose to use a very thin potential barrier called a delta barrier. In time-independent studies, this barrier provided results that captured many of the features of tunneling when more realistic barriers were used. We think that when the molecule hits the delta barrier, there's a chance that one of the molecules passes the barrier, but the other is reflected by it, and hence the molecule breaks up. However, there is still an attractive force drawing the molecules toward each other, so the atom that passed the barrier may be drawn back toward the atom that remained behind the barrier and eventually recombine with it.

This leads into another surprising result, one that is not considered in time-independent studies:

It is also found that, in addition to reflecting and transmitting, the molecule may also temporarily straddle the potential barrier in an unbound state.

In other words, the molecule, upon contacting the barrier, stays near the barrier for a relatively long time. This is what happens when the scenario described in the last paragraph occurs repeatedly, only without the molecule recombining and entering into a bound state. Straddling, as we called it, does not occur for a molecule in the bound state. In order for a molecule to break up, it needs energy. This energy comes from the initial kinetic energy of the molecule. Straddling occurs when the energy needed to break up the molecule is nearly the same as the kinetic energy of the molecule, so that when the molecule breaks up, the atoms don't have very much kinetic energy left. Again, this is a bit of an oversimplification, but it captures the main physical features of what's going on.

In the second study, we consider the case of a molecule incident in the bound state upon a step potential with energy less than the step. We show that in the limit where the binding energy e0 approaches zero and the step potential V0 goes to infinity, the molecule cannot remain in a bound state if the center of mass gets closer to the step than an arbitrarily large distance x0 which increases as the magnitude of e0 decreases, as V0 increases, or both. We also show that, for e0→0- and V0→∞, if the molecule is incident in the bound state, it is reflected in the bound state with probability equal to unity, when the center of mass reaches the reflection distance x0. We verify that the unbound states exhibit the expected physical behavior. We discuss some surprising results.

The second study, unlike the first, was entirely analytical, i.e. pen and paper mathematics, with no computers needed. What we considered was the case of a molecule that was extremely weakly bound incident upon a "hard wall" potential barrier, that is a potential barrier that was very long and very high. The binding energy is the term e0 referred to above; the term V0 refers to the energy "height" of the potential barrier. We considered this case, initially, as a simple test of our calculations. It turned out that this "simple" case was actually very hard, and yielded very counter intuitive results, as I'll explain below.

Intuitively, what we expected to happen for the weakly bound molecule to come close to the barrier and break up, with the atoms reflected away from the wall. What we found instead is that there is a distance, x0, from the wall within which the molecule cannot remain in the bound state. The distance x0 grows larger the more weakly bound the molecule is. Furthermore, we found that the probability of the molecule being reflected in the bound state approaches 100% in the case of extremely weak binding and extremely large potential barrier height. Taken together, this means that a weakly bound molecule, coming towards the hard wall potential barrier from a very long ways away, comes to a within a distance x0 from the barrier, and is then reflected away from the wall in the bound state. To get a bit of an idea of how weird this is, imagine throwing a brittle champagne at a brick wall. You'd expect it to hit the wall and shatter, with shards of glass boucing back. If the glass behaved like a weakly bound molecule, what would happen instead is that the glass comes within 50 feet of the wall and bounces back, intact. A champagne glass is more than a little bit different from a diatomic, homonuclear molecule, I know, but you get the idea.

Connections between our results and investigations done in cold atoms, excitons, Cooper pairs, and Rydberg atoms are discussed.

Apart from the sheer difficulty of the calculations, another problem with the study of molecular tunneling is in connecting it to real world applications. Direct experimental applications don't yet exist. However, connections can be drawn to many real world systems. Rydberg atoms, for instance, can be modelled pair of weakly bound particles, i.e. a very high energy electon and an atomic nucleus + lower energy electons. Rydberg atoms can also combine to form very weakly bound molecules. Collisions of Rydberg atoms with the surfaces of certain materials has been investigated. This scenario is akin to a weakly bound molecule incident upon a hard wall.

The tunneling of other composite particle objects, like excitons and Cooper pairs, can also be studied and are a subject of research interest. Cooper pairs are basically bound pairs of electrons which exist inside superconductors, and are indeed what make supercondutivity possible. Excitons are weird things that form inside of semiconductors and other materials. Basically, when an electron in such a material becomes excited, i.e. gains energy (by means of a photon collision, for example), it leaves an "electron hole," or absence, in whatever state it used to be it. This "hole," weirdly enough, behaves like another particle, and what's more, it can become bound the excited electron, forming an electon-hole "molecule" known as an exciton.

So, there you have it. I've summarized my crowning acheivement as a physicist, and with that out of the way, I'll get back to work on what really matters-- Sailor Moon: The Movie!