Particle in a Box

What does it mean to be a particle in a box? Is it some kind of existentialist metaphor, or is it a quandary we might find ourselves in from the Twilight Zone? No, even worse, it’s quantum mechanics.

More exactly, it’s a visualization thing in QM. We picture a particle in a box, and from that we can describe mathematically what is able to happen, given that scenario. So let’s call it “the particle-in-a-box visualization” of QM. It might be called “an interpretation,” but I am not here trying to argue for or against any of the standard opinions about QM. I’m just recounting how this metaphor is incorporated into QM.

But I do have my own agenda, which is to talk about “the box.” In part one of this essay, “On the Importance of Randomness,” I argued how thermodynamics takes random actions (heat energy) to be fundamental. Other forms of energy, as well as complex matter, are created by how thermal energy is captured within certain arrangements. Having energy makes a particle move randomly just from its own internal energy—it does not need to be causally pushed along in order to move randomly—but also, being made of energy enables particles to stick together so as to make these arrangements. Thus, what is fundamental is not so much the random actions alone but rather how the random actions, when occurring within certain arrangements, create complexity.

And so my agenda is to ask: Is QM the same way (since QM is the other fundamental science besides thermodynamics)?

And it turns out that, yes, QM already has a notion of its own called “a particle in a box.” If we take “the box” to be the special setup of circumstances inside of which the particle is moving, then we have the same metaphor as in thermodynamics.

For a concrete example, think of “the box” as being the rest of the atom, such as the nucleus and surrounding space, within which an electron is moving.

Then hold on tight. The derivations will show that even quantization itself is a quality that exists because of how things are happening “in a box.”

By focusing on derivations, I am here being consistent with other posts that I have been making regarding the philosophy of science. We can get one impression of science if all we do is look at the results, and we can get another impression if we keep in mind how we (or nature) have set things up to get those results.

boundary conditions

Quantization does, in fact, occur in classical physics. Say that we take a rope and tie it between two trees. Then we grab the rope in the middle, depress it, and finally release it so that it wobbles up and down. Now, compared to just being a rope, the rope has additional features, such as how fast it vibrates up and down, and how much energy it has. And these new features can be related in equations, such as how its vibration frequency is connected to its energy.

My point is that the rope being tied at each end is what creates these additional features and their relationships. Without that larger setup, the rope is just a rope.

But of course I am describing the famous “standing wave.” (Notice that it isn’t going anywhere, as in a “travelling wave”). And what we have learned, classically, from the standing wave is that one of its new features, created from how its two ends cannot move, is that it can only vibrate at certain frequencies. Its energy levels are quantized, meaning that only certain amounts of energy can fit with the conditions that the ends of the rope cannot move. And that is evident in the more common examples of standing waves, which are the strings on guitars or violins. The outcome is musical harmonics where only certain overtones are allowed. The limitation that no movement can happen at the ends of the strings—the boundary conditions—is what creates the outcomes of quantized energy levels.

And as is well-known, the Schrodinger equation, at the heart of QM, has the mathematical form of a standing wave.

It is sometimes argued that a standing wave is actually a superposition of two travelling waves, one moving left to right, and the other going right to left. But whether that is a handy way of re-imagining it or not, my point is that, either way, the rope’s boundary conditions—its necessarily being stationary at each end—is what creates its additional features, including quantization.

And now we can extend that to three dimensions, so that, instead of a rope tied at each end, we can have a particle roaming in a box. The particle is not tied to the walls, but even so, there are places where it cannot move into. There are boundary conditions, such as how an electron cannot move into the place in an atom where the nucleus resides. And as with the classical standing wave, the boundary conditions create quantization. They create how the electron can have only certain energy levels if it is to fit that setup.

And that is what I started out to show, that the larger setup has a role in making what can happen, even in QM, as it also occurs in thermodynamics.

I will return to more about the derivations of quantization later in this essay. But first, I want to address two possible questions, to complete the picture. Why should we think of the electron (or other quantum entity) as moving randomly? And what are the advantages of doing so? (But now, regarding that, I do not mean to imply that this is settled science, so perhaps I should call it an “interpretation,” after all).

visualization

It helps to start by looking at why Max Born first proposed that the Schrodinger equation must be describing probabilities, not a literal wave. A few people, notably Einstein and Schrodinger himself, objected that they did not share Born’s intuition—they wanted to maintain the old mechanical philosophy that causality, not randomness, is paramount—so it is worth recounting the problems which Born’s approach solves. 

The initial problem with taking the Schrodinger equation as literally describing a wave was that waves spread out and dissipate. So, if the most fundamental thing in the Universe is a wave, that means that the Universe should have long ago dissipated into nothingness. Yet the Universe is still here, so the problem is to explain why. Also, taking the Schrodinger equation as literally describing a wave does not explain the direct observations that subatomic entities do things in the manner of point-like particles, not waves, such as leave trails in cloud chambers

Born’s solution was that an electron moves as a particle, but the chances of finding it in a certain position is given by a formula that happens to look like a wave. But what does that mean (other than that the Universe has been saved from annihilation)?

It goes back to that other deep distinction between quantum and Newtonian mechanics (besides quantization) which is that, in classical physics we can predict exactly where an entity will end up, given its initial conditions, but in QM we cannot. In practice, we can only give the odds of where it will be later on. So, that is the observation that needs to be explained. And suddenly, it makes sense to speak of the probabilities of it being in one location or another. If we cannot predict exactly where it will end up, then we can still give the odds of where it might be.

Thus Born talks of probabilities.

And my point is to ask: But how can we speak of the odds of it being in a certain position unless we include talk about the larger setup that it is in? How can it have a position unless we factor in that which it has a position “in”?

So we should not be surprised to find that QM ends up talking about a particle in a box. And the particle is moving randomly, at least in the sense that it is unpredictable where it will end up. It is just that now we can give the odds of where it will be in reference to the box.

Yet just because the particle moves randomly, that does not mean that it doesn’t have features of its own and that we can relate these features in equations. (Arguably, these features, such as spin and angular momentum, might even come into existence as expansive new features from being within the boundary conditions of the box, the way that a rope tied to two trees now has the new feature of vibrating up and down). In any case, we can write equations relating these features. But if those equations happen to look like the equations of a wave, that does not necessarily mean that the particle is therefore itself literally a wave.

The mathematical function describing the relationships among these features need not be real—only the physically measurable quantities must be real—and we can describe their relationships using any kind of mathematical chicanery that we like. Indeed, imaginary numbers appear in the Schrodinger equation. 

So Born can say that the odds of finding an electron in a certain position is given by a formula that happens to look like a wave without it necessarily being a wave. It is like how F = ma and C = πd both take the same mathematical form a = bc, but that doesn’t mean that circles necessarily are really about force and acceleration. And likewise, the Schrodinger equation is not necessarily describing a wave just from sharing a math format with a wave.

advantages

And then there turns out to be other advantages to treating a QM problem as a particle moving randomly within a box.

For starters, that makes it easier to visualize how an electron can be on either side of an atom but never in-between. That is less mindboggling if we understand that we are only talking about the odds of random events happening, not the behavior of a literal wave.

Some people object to the notion of finding probabilities at the most basic level of physics. They find probabilities to be too esoteric for what is fundamental; they want something more concrete to be what is ultimately real. But particles moving randomly are very concrete. It is just our ability to know their location that is probabilistic.

And there is yet another problem with having waves as the basic stuff of the Universe. That is that waves are “not all here” at any given moment. To tell it’s a wave, we have to look over time, to appreciate how it is going back and forth. At any given moment, most of what it “is” is out of the immediate picture. So a wave lacks immediacy.

Born was saving physics from being the study of what is fundamentally not all here at any given moment. And do people wanting concreteness in ultimate reality really think that non-immediacy is the answer?

But a particle moving randomly is indeed all present at any immediate moment (it’s just that we don’t know where it is)—so it has a strong claim on being “real.” The cost of being random is not to our sense of realness, but just to our sense of causality, since random movements are not causally predictable. Born was making a trade-off. He was giving up micro-scale causality and mechanical philosophy in order to salvage immediacy (all-here-ness) in what is fundamental.

Yet if we have studied thermodynamics first (before QM), then we know that thermodynamics has already abandoned a mechanical philosophy by treating heat energy (randomness) as basic. So for Born to do the same with QM was only to make the two theories be compatible in their core assumptions.

orbitals

But now I want to turn to two ways in which other sciences actually use QM. To the extent that what a thing even “is” is defined by how it works operationally, we can find some meanings that way, too. We might at first suspect that being confronted with randomness indicates that we have encountered an obstacle (the randomness) that keeps us from making practical applications. (At least superficially, it would seem that randomness means useless unpredictability). But no. As with thermodynamics, it turns out that random actions, if within certain setups, have remarkable utility. And QM, with its “box,” is no exception.

QM is well-known for stipulating that all we can do is find the probability of an event happening. And that is true compared to a Newtonian event. But there is actually more that we can do with randomness than just give the odds, when we factor in the set of circumstances.

First of all, it is important to realize that even classically we can still make equations describing random actions in ways other than just by giving the odds. A straightforward example is how a gas (air), which by definition is composed of molecules moving randomly, still has a pressure, if the randomly moving molecules are within a closed container. Also, they have a temperature. So we can still make equations, the gas laws, relating pressure and temperature, even though doing so is regarding what is moving randomly. These equations are not about finding the odds of what an individual molecule is doing but about relating the expansive features (such as pressure) that many molecules have altogether.

So in QM, we can do the same thing. We can look for features that can be found in many random events altogether and then relate these features to each other and to observable outcomes. Again, it works because it is all happening within a specific setup which contributes to making these expansive features (the way that being in a container makes pressure).

Even random events can have relationships, if we consider them in bulk and as occurring within certain arrangements. And what has relationships can be described with equations.

We do that all the time in chemistry. Another example is equilibrium. That is where zillions of random molecular actions, if contained together, create a product at the same rate as they fall back apart and undo that product. We do not need a sense of causality to understand that, since they randomly—unpredictably—combine and fall apart; we can’t predict what any individual molecule will do. Yet we can describe the overall process mathematically, with equilibrium constants, and use that to make predictions and to discern still other features, such as about what else affects the equilibrium (such as Le Chatelier’s principle). Chemistry is full of variations on this theme—dissociation constants, rate constants, feedback of various forms, concentration dependencies—and so to find it happening in QM is just more of the same.

We are used to dealing mathematically with randomness in ways besides just finding the odds of individual outcomes. We are interested in the macro-scale (observable) outcomes.

So a way to do something comparable in QM is with atomic orbitals. Instead of considering just one individual outcome (one place where the electron can be), what happens if we consider every possible outcome altogether (all the places where it can be in “the box”)? What we can do is plot that on a graph—we can put a dot on a piece of paper or on a computer screen for every possible place that an electron can end up—and then we can see what kind of picture that makes. It ends up making a shape, called an “orbital,” such as a sphere or a dumbbell. And then we can make predictions and explanations based on understanding these orbitals in what is called “molecular orbital theory” (or MO theory).

A similar approach is Feynman’s quantum electrodynamics, where the electron is a particle “free to move however it likes,” and then all the possible paths are summed in his “path integrals.” But what defines these paths except for being in some kind of a setup (showing that “the box” can be very large, indeed, not just the rest of a molecule)?

These are further instances of how additional expansive features (the orbitals, the paths) can exist because of the setup, and because of considering many instances instead of just one, and of how then we can relate these extra features to each other to predict observable outcomes. It works even if the new expansive features do not exist in the individual all by itself. For instance, an individual molecule cannot be in equilibrium all by itself. And yet that enables predictions of macro-scale outcomes. (And it further illustrates another point that I sometimes make, which is that individual traits are not always intrinsic—they are not always about “the thing-in-itself,” as Plato and Kant would have it—since, instead, traits can depend on what arrangements the individual is within).

quantization

Then a second way of understanding the Schrodinger equation in operation is to consider how it is used mathematically to make further derivations. Here, I will only sketch the process (at least as it works in chemistry) to make my point.

There are infinitely many possible solutions to the Schrodinger equation. So how do we get the answer to the particular problem that we’re trying to solve? 

In practice, there are three steps to using the equation. First, we describe the boundary conditions of the setup under scrutiny. (We tell where the electron cannot go, where there must be a zero in the math). Then the second step is to solve the equation using these boundary conditions to find what energy levels the equation yields for that setup. And then three, we use these values for energy to calculate the observable (macro-scale) properties of the system.

The simplest assumption, for ease of doing math, is to treat the boundary conditions as making a three-dimensional rectangle. From that, we can calculate, for instance, such things as the volume. That approach has wide applications, although, obviously, it is just an approximation. But that is how we arrive at the phrase “a particle in a box

Again, my point is that “the box”—the wider setup—is integrally incorporated into QM, just as it is in thermodynamics. The box (the places where the electron cannot enter) makes it so that only certain energy levels fit with having that box, so we arrive at the quantization of energy levels. We can’t skip the box and still have a quantum system.

Even Planck, in his original derivation of his famous constant establishing quantization, did so in terms of standing waves within the boundary conditions of being in a blackbody. (We now speak of blackbody “radiation,” but Planck originally worked in terms of standing waves).

And we can see why all of that makes it difficult to see the Schrodinger equation as describing a literal travelling wave, since a quantum entity cannot leave its boundaries and still be a quantum system. The boundaries are what make it a quantum system.

A quantum entity does move, but it moves randomly, not as a literal wave.

Yet then again, could the act of a measurement destroy the boundary conditions that are making the system be quantized in the first place? Is that what it means to have “a wave collapse”?

problem of measurement

Yes, I think it is even possible to understand the so-called “problem of measurement” (or “wave collapse”) in terms of a particle in a box. The problem is to explain the role of the observer, since a quantum system seems to possess a set of possible outcomes, and the act of measuring it makes one of those outcomes occur; but it is unpredictable which outcome it will be. The equation that describes those possible outcomes breaks down—the equation that looks like a standing wave collapses—as one outcome actually takes place because of the measurement. 

Heisenberg describes it, in his book on philosophy, as the quantum system exhibiting “potential” —the equation displays the potential outcomes—and the measurement actualizes one of those possibilities. But people who insist on a mechanical philosophy, where one immediate (or “real”) event leads successively to the next, find it queasy to believe the notion that potential could be fundamental. They find it to be too abstract—they want an explanation that is more concrete and immediately happening than just potential—to describe what is most basic to the Universe.

But with a particle in a box, it is not about “potential” per se, since the particles are concrete. It is just that, by moving randomly, the particles have the potential to appear in different places.

To my thinking, these objections to potential are missing the role of “the box” in the particle in a box visualizations of QM. As we have seen, the quantum system only has its quantum features because of the setup of boundary conditions, so when the act of a measurement disturbs those boundary conditions (think of another molecule bumping into the one in question), the system collapses; the set of possibilities becomes one actual outcome. Since the particles are moving randomly—with no pattern—it is impossible to predict which of the potential outcomes will be actualized. Prior to the measurement (prior to the collision), all we can do is describe the various ways that it might turn out.

Then altogether, a thermodynamics-based interpretation of QM (one which takes randomness as fundamental) might look like this:

The boundary conditions put an “n” in the equations which stands for a positive integer, creating the quantization in how only whole multiples of n are allowed for a final result. With music, we can think of many waves in a vibrating string as being superimposed, as the waves move back and forth between the stationary ends of the string at different frequencies, to create n number of simultaneous overtones. But with energy levels, it is more likely that nature settles on just one multiple of n at a time, not to be in all of them simultaneously, and that it can jump back and forth among the levels (a principle used in lasers). Thus, the other energy levels exist originally only as potential outcomes. That is especially so if the particles are moving randomly rather than vibrating periodically as in harmonics. (The periodic behavior is perfectly predictable, unlike QM behavior). In any case, the “waves” cannot be superimposed unless they are in some way bounded by the larger setup, which is my point. And breaking those bounds can accordingly result in wave collapse or the end of superposition or of potentially having other outcomes.

If we try to say that the system doesn’t actually collapse, that it just goes somewhere else into the environment, then we are forgetting that the quantum qualities only exist by virtue of how the particle cannot cross certain boundaries. It ceases to be a quantum system if it “leaves the box” by going into the environment. (But there is also the issue of whether “information”—even if not the particle itself—can “leak” or “smell” or “tunnel” through the boundaries without the system collapsing, perhaps via entanglement. I will leave that question to the physicists).

The questions that I am addressing are: How does the Universe build complexity? Can random actions be fundamental and still have the world work? Andy yes, they can. Both thermodynamics and QM show that, when the random actions occur within arrangements or boundaries, they make the macro-scale outcomes that we observe.

broader scheme of things

Well, I wanted to get into the issues suggested at the beginning of this piece, about a particle in a box being a Twilight Zone style metaphor for daily life: a person within a culture, society, or a family. But I see that I am out of space. That will have to wait for another essay.

IMAGE: Please forgive my drawing on restaurant napkins.