Category Archives: Philosophy of Physics

Why quantum mechanics does not say a particle can be in two places at once

This post is about understanding the difference between a mathematical model and the physical system that is being modeled. Confusion between the model and the physical system is especially prevalent in quantum physics and leads to statements such as:

  • A particle can be in two places at once
  • An object is spread out in space until you measure it
  • Sometimes it’s a particle and sometimes it’s a wave

This post consists of three parts. First we’ll discuss a very simple example of a mathematical model vs a physical system: a ball being thrown into the air. Then we’ll go on to a more interesting example: the heat equation.

Finally, we’ll tackle Schroedinger’s equation. Without worrying about how to solve the equation, we will simply discuss the mathematical function (the model) that the equation refers to. Then we’ll try to see how that function relates to a physical system (sometimes called a “particle”). Hopefully this will clarify why mis-statements like the ones above are so common.

A ball thrown into the air

If you throw a ball straight up into the air, its path (ignoring air resistance) can be calculated using Newton’s “2nd law” F=ma. The calculation itself is not important for our purposes, but if we plot the resulting height of the ball for the time it is in the air we get a parabola as in the following diagram:


However, the parabola that appears in the diagram does not appear anywhere in physical space. This sometimes confuses people the first time they see a picture like this. If you don’t notice that the horizontal axis is time it’s easy to imagine that the ball is being thrown from one place to another. In fact, if you do throw a ball from one place to another, its path will be a parabola. But that’s not what we see in the diagram above. That ball is simply going straight up and down in physical space.

This is a very simple example of the difference between a mathematical model (the parabola above) and the physical system it represents (a ball going straight up and down).

The Heat Equation

Another (more interesting) example of the model vs the reality is provided by the heat equation. The simplest form of the heat equation is:

    \[\frac{\partial}{\partial t}T=k\frac{\partial^2}{\partial x^2}T\]

In this form, the heat equation could represent the temperature inside a narrow tube. The idea is that (in an ideal situation) the tube could be so perfectly insulated that heat only flows out the two ends.

The function T

The function T in the equation above represents the temperature at each point in the tube. When we solve the heat equation, it tells us the value of T for any time we choose. We then know the temperature at any point in the tube at that time.

We can imagine the tube lying along the x axis and we could then plot the temperature funcion T, for a given time on the y axis, like so:


Now, even though this is a very common way to visualize the heat equation, if you think about it, it’s kind of a strange mixture of the mathematical model (the blue curve, T) and the physical system that it represents (the gray wire along the x axis). The actual temperature is not a curve hovering above the wire. What the temperature really is, is the motion of the molecules inside the wire.

But there is a straightforward relationship between the blue line and the temperature. Where the line is higher, the molecules are moving faster. So as long as you keep this in mind, it’s easy to keep the two concepts separated and there’s no harm in drawing it all together.

Schroedinger’s Equation

Now we come to Schroedinger’s equation, the fundamental equation used for basic non-relativistic quantum mechanics. This is the equation, in its simplest form:

    \[i\hbar\frac{\partial}{\partial t} \mbox{\Large$\psi$} =-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \mbox{\Large$\psi$}\]

This is the equation that would be used to predict the location of a single particle’s position along the x-axis, if that particle is not being subjected to any forces. We’re not going to solve the equation. We’re simply going to assume that we already have a solution and talk about what that solution means.

The function \psi

Just as the function T was the solution to the heat equation, the function \psi is the solution to Schroedinger’s equation. But \psi (sometimes called the “wave function” or the “state vector”) is a complex function. That means its values consist of both a real part and an imaginary part. (The imaginary part is a multiple of i, the square root of -1.)

Since \psi is complex, we can’t simply plot its values along the x-axis as we did with T. There are various ways that we can attempt to visualize \psi. For example we can plot it in the complex plane. This means we plot the complex values on a real axis vs an imaginary axis, like so:


If we do it this way, then the wave function of this particular particle looks sort of like a spiral.

Another common way to visualize a complex function is to draw the real and imaginary parts separately, as if they were both real, but plot them in different colors. If we take the same wave function above, and plot the real part in blue and the imaginary part in red (both on the real plane) then we get this picture:


If we do it this way, it looks kind of like two overlapping “waves.”

So what is \psi really? Is it a spiral, is it waves, or what? The answer of course is that it is neither of these things. The wave function \psi exists in an abstract mathematical space. We can try to picture it in various ways, but it is not really any of those pictures.

The relationship between \psi and the position of the particle

Now let’s talk about the relationship between the mathematical function and the physical system. In our first two examples, the ball and the heat equation, we understood from the beginning what the physical system “really was.” We can see the ball going up and down, and we can see how the height of the real ball corresponds to the height of the parabola at a particular time.

With the heat equation, although we can’t actually see the molecules moving around, we can easily form a picture in our minds of what heat consists of in the tube, and we can understand the relationship between the height of the function T in the diagram, and how fast those molecules are moving around in the tube lying along the x-axis.

But the relationship between the wave function \psi and our idea of “where a particle is” along the x-axis is much more complicated.

First of all, we don’t use \psi directly, but rather we first apply an operation called the modulus squared. This yields another function |\psi|^2 which is actually the probability density function associated with finding the particle at a particular place on the axis.

|\psi|^2 can be drawn over the x-axis just like we drew the heat function T. Here’s a picture of |\psi|^2 for the same particle above:


The relationship between the probability density function and the position of the particle is that the area under the curve between two positions on the x-axis gives the probability of finding the particle between those two positions.


So let’s break that down:

  1. We are about to look for the particle.
  2. We want to know the odds that we’ll find it between point a and point b.
  3. We can find that probability by calculating the area under the curve between those two points.

In case you’re curious, this calculation is done with the integral:

    \[\int_a^b |\psi|^2 dx\]

We don’t have a “picture” of the particle!

As you can see, the relationship between a particle’s wave function and its position is very different, and much more complicated, than the relationship between the mathematical model and the physical system in our other examples. But the main reason for the confusion between the model and the reality doesn’t (mostly) arise from that fact.

The main issue is that in quantum physics we simply don’t have a “picture” of the particle’s position. When we actually do a measurement, and we see something — say a detector going off — then we typically say that we “saw a particle at position so-and-so.” But, in general, all quantum mechanics gives us is a recipe for calculating the probabilities of these outcomes. It doesn’t say anything about “positions” beyond that.

In fact, there is no universal agreement as to whether there is actually such a thing as a “particle” at the most fundamental level. Some take the position that there are only “quantum fields” and that what we call “particles” are simply excitations in those fields, which can be detected in certain ways under certain circumstances.

So what leads people to say things like the three statements at the beginning of this article? I can only speculate of course, but here are my thoughts.

First of all, they may just be repeating something that they heard or read. This is probably the most likely scenario 🙂

Another possibility is that someone who knows about quantum mechanics, and understands Schroedinger’s equation perfectly well, simply gets confused between the model and the reality. Since we basically don’t know what the “reality” is, this is understandable!

The wave function, and especially the probability density function, do kind of look like something that is “spread out in space.” Furthermore, if you’ve got a picture of the wave function in your head, and the you do a measurement, you could be forgiven for simplifying this mentally, to: “It’s a wave somewhere out there, but then when I measure it, it snaps into position as a particle.”

But even so, why is it that people specifically giving lectures on quantum physics — people who are teaching about the subject — lapse into this kind of language?

I think maybe this is what happens. And it most often happens in “popular” physics book and lectures. You come to the point in the lecture where you have to say something. The “something” that is really true is a purely abstract mathematical statement. But you’re pretty sure your audience won’t understand that, and doesn’t want to hear it. So you fall back on the only “picture” you have, which is a picture of a wave (or probability density) function and you just talk about it as if it were the physical reality.

You probably figure that it will at least give people some kind of notion of what’s going on. Or, at least it will convey a feeling for how unintuitive quantum physics can be. Or, at least it will sound cool …

However, in my view at least, it actually conveys a mistaken impression of “what quantum physics says.”

Final Note

OK, so I get where “spread out in space” and “sometimes a wave” come from. But I still don’t see anything that looks like a particle in “two places at once.”

Right. So the wave function above was for a very simple situation. Basically what’s called a “free particle.” In a more complicated scenario (but still for a single particle) you can ultimately end up with a probability density function that looks something like this:


As you can see, there is a pretty strong probability that an experiment will detect the particle either around point a or around point b. The probability is actually non-zero all along the line, but it’s strongly peaked near those two locations. This is where the “two places at once” notion comes from.


Filed under Philosophy of Physics, Quantum Theory

Is Quantum Mechanics/Theory Better Understood as Science or Magic?

Two of the over eleven definitions of the word “understand” in The Merriam-Webster Dictionary.

  • #2 : to accept as a fact or truth or regard as plausible without utter certainty <we understand that he is returning from abroad>
  • #5 : to achieve a grasp of the nature, significance, or explanation of something

Summary: The following brief paper proposes that, of all the fields of scientific study, quantum mechanics has failed the most profoundly in being able to explain the nature of the phenomena it seeks to understand. And, while this is not a failure of the scientists investigating the world of the very, very small, it speaks to the quantum realm having more in common with the world of Harry Potter than it does of our collective understanding of “real things.”

Introduction: Two weeks ago this Sunday, a group of us had a discussion concerning whether we (or anyone, for that matter) really understands quantum mechanics.

On the pro-side of the argument was the statement, attributed to Richard Feynman, that “if you can build something, then you understand it.” Therefore, because we build all sorts of functional machines that use quantum mechanics as the basis of their design, then it must follow that we do, in fact, “understand” quantum theory and the quantum world. Another way of stating this (as I understand the argument) is that, by understanding the mathematics of how a thing functions, one understands, in some fundamental way, the thing itself.

The con-side of the argument (this author’s position), is that “understanding” anything dealing with the quantum world (see definition #5, above) is, for now at least, beyond our grasp. For example, to say some object has a “wave-particle duality” means that we know that this something acts in a certain way, but that we have no clear comprehension of, or agreement as to the how or why of the phenomena. Simply saying that something is the way it is because, well, dammit, that’s just the way it is, is an assertion, not a scientific explanation.

Another way of stating this position is that, like the classic cartoon in which two scientists are standing at a green-board, studying a set of equations, we need a better understanding of what exactly the miracle of “Step Two” is. And, so far as I understand quantum mechanics, a lot of people have many, many interesting ideas of what that “miracle” is, but very few agree…

The Nature of Science:  Prior to the discovery of how tectonic plates move, Alfred Wegener’s ideas concerning continental drift were considered by the scientific community as so much misguided naïveté (to be generous). Only with the final acceptance of a logical (and demonstrable) mechanism for the movement of massive sections of the earth’s crust was the theory finally brought within the realm of “real science.” (see definition #5).

The acceptance of evolution as a scientific “fact” is even more instructive. With the publication of The Origin of the Species in 1859, Darwin proposed a radically new theory (sort of) on how species develop. But it wasn’t until others linked Mendel’s work on heredity with Darwin’s careful observations and hypotheses that the debates ended and evolution became the cornerstone of much of modern science.

And what was true for continental drift and evolution is true of almost all the sciences, from astronomy and archaeology, through classical physics to volcanology and zoology. In the world of science, basic principles are discovered, investigated and expanded upon. But even more importantly, there is a logical, sequential process in which each new discovery is built on a deep knowledge and understanding of what came before. Even when discussing Thomas Kuhn’s idea of paradigm shifts, the “new” scientific paradigm must describe the logical linkages as well as, if not better than, the previous one.

The Nature of Magic:  For the purposes of this discussion, let us separate a magician’s parlor tricks from “real magic,” and assume that “real magic,” as described in fantasy novels (Harry Potter, et. al.) , somehow actually works. Not any one set of “magical rules” from any one novel, of course, but the range of what is popularly understood of as “magic.”

In the world of “real magic,” the basic law is that, if I do THIS, then THAT will happen… If I say the correct magic words, in just the right order, with just the right pitch, then some outcome I desire will occur. Or, if I wave my hands in the proper pattern, with the appropriate scowl, then something amazing will appear.

If asked to describe HOW the magic happens, the magician may describe how things below are reflected in things above, or visa-versa. Or he might tell you how a blood sacrifice allows him to pull power from the hidden gods… or she might reluctantly explain how reciting the true names of tree nymphs binds them to her, and gives her their power.

But what you will not hear is a description of how magic works, based on any known physical principles. There will be no logical train of material processes/properties that, when fully articulated, will lead any layman from an understanding of how the mundane world works to one in which “real magic” functions.

The Conflict Between Science and Magic: On the other hand, in science, if we do THIS, and THAT happens… the scientist can (and probably will) give you a step-by-step description of how one physical process or property leads to the next, and how the whole chain of events produces the outcome. And this process can be (for the most part) followed and understood by any reasonably intelligent listener.

Mostly…  The one field of study where a logical, sequential explanation of what is happening breaks down is quantum theory (and, okay, maybe cosmology). So, to the question of what is the difference between science and magic? I propose that the most important difference is that, while one can understand THAT magic might work, one must be able to understand HOW something works for it to be considered a “real science.” There is no science I know of where the scientist, in attempting to describe her field of study, says… yeah, well, we have absolutely no idea how it works. It just does, and that’s good enough for me. Again, this is true of all the sciences, other than in the realm of the very, very tiny (and of the very, very big).

As one enters the world of quantum theory, there is a certain point, as one travels down the scale from the macro world to the micro, that the journey slowly turns from the study of science into a discussion of magic. From the macro world of this is HOW and WHY something happens/works to the micro, where it just DOES, and we’re still trying to figure out that whole, pesky HOW and WHY thing.

Describing vs. Explaining: Math does an unparalleled job of describing the quantum world. In fact, it is one of the only ways to deeply explore that environment. But arriving at the formula E = MC2 does not explain why E should be equal to MC2, only that it is. Why is it not equal to the cube of C? Well, because the math leads you to the square, not the cube. When people ask why; ask to have something explained, the answer they are looking for is a description of the basic principles involved, and how those principles lead to the outcome.

In quantum mechanics, the physicist is almost always forced to say “This is what happens. We have proof that this happens. But let me give you my personal belief as to WHY it happens.” The physicist can describe in mind-numbing detail what happens. She can discuss with other physicists the math of what happens. But when required to explain why this thing happens to a non-mathematician, she is thrown back on using metaphors and similes.

So, for the sake of further discussion, let us pull apart the meaning of the word “understand” into two separate components. The first will be UD; understanding that comes from being able to describe a thing or system. The second will be UE: an understanding of a thing or system that allows for explaining how and why it works. Usually, in the macro, mundane world, the two meanings of the word are mutually interdependent. But the more one studies quantum theory (and to some degree cosmology), the more one relies on UD, and abandons UE.

Post Script/Math: Math is not fundamental when trying to help someone understand how evolution works, or when the first humans came to America. It’s not even really necessary when describing the basic principles of bridge building or cathedral erecting (neither the Roman’s nor the medieval French used advanced mathematics). In almost all cases, math is supportive of, but not essential to, an understanding of a scientific or engineering subject.

If one wants to have a truly deep understand of engineering, then yes, math is absolutely necessary. But in those topics normally thought of as “scientific,” math is, at best, the junior partner.

Last Thoughts: There are so many explanations and descriptions of what is or might be going on at the quantum level… pilot waves, standing waves, multiple dimensions (4, 6, 8, 10 or 26), strings instead of points, tunneling, entanglement… the interested lay-person is often left wondering what physicists really know, and whether or not they telling the general public a collection of JUST SO stories. And, in this field at least, the distance between knowing what is happening, and understanding why it is happening continues to be a seemingly unbridgeable gulf. Not forever, certainly, but for now.

Pick up almost any issue of Scientific American or Discover magazine, and you will read articles on the latest discoveries in archaeology, astronomy, meteorology or paleontology, but only descriptions of the latest theories concerning the quantum world.

Summary: Arthur C. Clarke’s Third Law states that “any sufficiently advanced technology is indistinguishable from magic.”

I humbly concur. I believe our understanding of the quantum world will remain more like magic than science until new theories are developed and tested that give one, and only one explanation for why “wavicles” have their dual nature, how electrons can tunnel and if entangled particles are really one thing or two.

Until then, it’s just magic, all the way down.




Filed under Philosophy of Physics