Nature could be said to be constructed out an immense number of physical processes… indeed, that’s almost the definition of “physics”. But what makes a physical process a measurement? And once we understand that, what makes a measurement in quantum physics, a fraught topic, different from measurements that we typically perform as teenagers in a grade school science class?
We could have a long debate about this. But for now I prefer to just give examples that illustrate some key features of measurements, and to focus attention on perhaps the simplest intuitive measurement device… one that we’ll explore further and put to use in many interesting examples of quantum physics.
Measurements and Devices
We typically think of measurements as something that humans do. But not all measurements are human artifice. A small fraction of physical processes are natural measurements, occuring without human intervention. What distinguishes a measurement from some other garden variety process?
A central element of a measurement is a device, natural or artificial, simple or complicated, that records some aspect of a physical process semi-permanently, so that this record can be read out after the process is over, at least for a little while.
For example, the Earth itself can serve as a measurement device. Meteor Crater in Arizona, USA is the record of a crude measurement of the size, energy and speed of a large rock, as well of how long ago it impacted Earth’s surface. No human set out to make the measurement, but the crater’s details are just as revealing as any human experiment. It’s true that to appreciate and understand this measurement fully requires work by humans: theoretical calculations and additional measurements. But still, it’s the Earth that recorded the event and stored the data, as any measurement device should.
The Earth has served as a measurement device in many other ways: its fossils have recorded its life forms, its sedimentary rocks have recorded the presence of its ancient seas, and a layer of iridium and shocked quartz have provided a record of the giant meteor that killed off the dinosaurs (excepting birds) along with many other species. The data from those measurements sat for many millions of years, unknown until human scientists began reading it out.
I’m being superficial here, skipping over all sorts of subtle issues. For instance, when does a measurement start, and when is it over? For instance, did the measurement of the rock that formed Meteor Crater start when the Earth and the future meteor were first created in the early days of the solar system, or only when the rock was within striking distance of our planet? Was it over when Meteor Crater had solidified, or was it complete when the first human measured its size and shape, or was it finished when humans first inferred the size of the rock that made the crater? I don’t want to dwell on these definitional questions today. The point I’m making here is that measurement has nothing intrinsically to do with human beings per se. It has to do with the ability to record a process in such a way that facts about that process can be extracted, long after the process is over.
The measurement device for any particular process has to satisfy some basic requirements.
- Pre-measurement metastability: The device must be fairly stable before the process occurs, so that it doesn’t react or change prematurely, but not so stable that it can’t change when the process takes place.
- Sensitivity: During the interaction between the device and whatever is being measured, the device needs to react or change in some substantial way that is predictable (at least in part).
- Post-measurement stability: The change to the device during the measurement has to be semi-permanent, long-lasting enough that there’s time to detect and interpret it.
- Interpretability: The change to the device has to be substantial and unambiguous enough that it can be used to extract information about what was measured.
Examples of Devices
A simple example: consider a small paper cup as a device for measuring the possible passage of a rubber ball. If the paper cup is sitting on a flat, horizontal table, it is reasonably stable and won’t go anywhere, barring a strong gust of wind. But if a rubber ball goes flying by and hits the cup, the cup will be knocked off the table… and thus the cup is very sensitive to the collision with the ball. The change is also stable and semi-permanent; once the cup is on the floor, it won’t spontaneously jump back up onto the table. And so, after setting a cup on a table in a windowless room near a squash court and returning days later, we can figure out from the position of the cup whether a rubber ball (or something similar) has passed close to the cup while we were away. Of course, this is a very crude measurement, but it captures the main idea.
Incidentally, such a measurement is sometimes referred to as “non-destructive”: the cup is so flimsy that its the effect of the cup on the ball is very limited, and so the ball continues onward almost unaffected. This is in contrast to the measurement of the rock that made Meteor Crater, which most certainly was “destructive” to the rock.
Yet even in this destructive event, all the criteria for a measurement are met. The Earth and its dry surface in Arizona are (and were) pretty stable over millennia, despite erosion. The Earth’s surface is very sensitive to a projectile fifty meters across and moving at ten or more kilometers per second; and the resulting deep, slowly-eroding crater represents a substantial, semi-permanent change that we can interpret roughly 50,000 years later.
In Figure 2 is a very simple and crude device designed to measure disturbances ranging from earthquakes to collisions. It consists of a ball sitting stationary within a dimple (a low spot) on a hill. It will remain there as long as it isn’t jostled — it is reasonably stable. But it is sensitive: if an earthquake occurs, or if something comes flying through the air and strikes the ball, it will pop out of the dimple. Then it will roll down the hill, never to return to the its original perch — thus leaving a long-lasting record of the process that disturbed it. We can later read the ball’s absence from the dimple, or its presence far off to the right, as evidence of some kind of violent disturbance, whereas if it remains in the dimple we may conclude that no such violent disturbance has occurred.
What about measurement devices in quantum physics? The needs are often the same; a measurement still requires a stable yet sensitive device that can respond to an interaction in a substantial, semi-permanent, interpretable way.
Today we’ll keep things very simple, and limit ourselves to a quantum version of Fig. 2, employed in the simplest of circumstances. But soon we’ll see that when measurements involve quantum physics, surprising and unfamiliar issues quickly arise.
An Simple Device for Quantum Measurement
Here’s an example of a suitable device, a sort of microscopic version of Fig. 2. Imagine a small ball of material, perhaps a few atoms wide, that is gently trapped in place by forces that are strong but not too strong. (These might be of the form of an ion trap or an atom trap; or we might even be speaking of a single atom incorporated into a much larger molecule. The details do not matter here.) This being quantum physics, the trap might not hold the ball in place forever, thanks to the process known as “tunneling“; but it can be arranged to stay in place long enough for our purposes.
If the ball is bumped by an atom or subatomic particle flying by at high speed, it may be knocked out of its trap, following which it will keep moving. So if we look in the trap and discover it empty, or if we find the ball far outside the trap, we will know that some energetic object must have passed through the trap. The ball’s location and motion record the existence of that passing object. (They also potentially record additional information, depending on how we set up the experiment, about the object’s motion-energy and its time of arrival.)
To appreciate a measurement involving quantum physics, it’s often best to first think through what happens in a pre-quantum version of the same scenario. Doing so gives us an opportunity to use two complementary views of the measurement: an intuitive one in physical space and more abstract one in the space of possibilities. This will help us interpret the quantum case, where an understanding of a measurement can only be achieved in the space of possibilities.
A Measurement in Pre-Quantum Physics
We’re going to imagine that an incoming projectile (which I’ll draw in purple) is moving along a straight line (which we’ll call the x-axis) and strikes the measuring device — the ball (which I’ll draw in blue) sitting inside its trap. To keep things simple enough to draw, I’ll assume that any collision that occurs will leave the ball and projectile still moving along the x-axis.
With these two objects restricted to a one-dimensional line, our space of possibilities will be two-dimensional, one dimension representing the possible positions x1 of the projectile, and the other representing the possible positions x2 of the ball. (If you are not at all familiar with the space of possibilities and how to think about it, I recommend you first read this article, which addresses the key ideas, and this article, which gives an example very much relevant to this post.)
Below in Fig. 4 is an animation showing what happens, from two viewpoints, as the projectile strikes the ball, allowing the ball’s motion to measure the passage of the projectile.
The first (at left) is the familiar viewpoint: what would happen before our eyes, in physical space, if these objects were big enough to see. The projectile moves to the right, with the ball stationary; a collision occurs, following which the projectile continues on the right, albeit a bit more slowly, and the ball, having popped out of its trap, moves off the the right.
The second viewpoint (at right) is not something we could see; it happens in the space of possibilities (or “configuration space,”) which we can see only in our minds. In this two-dimensional space, with axes that are the projectile’s and ball’s possible positions x1 and x2, the system — the combination of the projectile and ball — is at any moment sitting at one point. That point is indicated by a star; its location has as its x1 coordinate the projectile’s position at a moment in time, while its x2 coordinate is the ball’s position at that same moment in time.
The two animations are synchronized in time. I suggest you spend some time with the animation until it is clear to you what is happening.
- Initially, the star moves horizontally. This indicates that the value of x2 isn’t changing; the ball is stationary. Both x1 and x2 are initially negative, so the star is in the lower-left quadrant.
- Notice the diagonal line, at x1 = x2 ; if the system is on that line, a collision between the two objects is occurring, since they are at the same point. It is when the star reaches this line that the ball begins to move, and the star’s motion is correspondingly no longer horizontal.
- After the collision, both the projectile and ball move to the right, which means the values of x1 and x2 are both increasing. This in turn means that the star moves up and to the right following the collision, eventually reaching the upper-right quadrant where both x1 and x2 are positive.
By contrast, if the measurement device were switched off, so that the projectile and the ball could no longer interact, the projectile would just continue its constant motion to the right, unchanged, and the ball would remain at its initial location, as in Fig. 5. In the space of possibilities, the star would move to the right as the projectile’s position x1 steadily increases, while it would remain at the same vertical level because the ball’s position x2 is never changing.
The Same Measurement in Quantum Physics
Now, how do we describe the measurement in quantum physics? In general we cannot portray what happens in a quantum system using only physical space. Instead, our system of two objects is described by a single wave function, which is a function of the space of possibilities. That is, it is a function of x1 and x2, and also time, since it changes from moment to moment. [Important: the system is not described by two wave functions (i.e., one per object), and the single wave function of the system is not a function of physical space, with its coordinate x. There is one wave function, and it is a function of all possibilities.]
At each moment in time, and for each possible arrangement of the system — for each of the possible locations of the two objects, with the projectile having position x1 and the ball having position x2 — this function gives us a complex number Ψ(x1, x2; t). The absolute value squared of this number gives us the probability of the corresponding possibility — the probability that if we choose to measure the positions of the projectile and ball, we will find the projectile has position x1 and that the ball has position x2.
What I’m going to do now is plot for you this wave function, using a 3d plot, where two of the axes are x1 and x2 and the third axis is the absolute value of Ψ(x1, x2; t). [Not its square, though the difference doesn’t matter much here.] The colors give the argument (or “phase”) of the complex number Ψ(x1, x2; t). As suggested by recent plots where we looked at wave functions for a single particle, the flow of the color bands often conveys the motion of the system across the space of possibilities; you’ll see this in the patterns below.
Going in the reverse order from above, let’s first look at the quantum wave function corresponding to Fig. 5, when no measurement takes place and the projectile passes by the ball unimpeded. You can see that the peak in the wave function, telling us most probable values for the results of measurements of x1 and x2, if carried out at a specific time t, moves along roughly the same path as the star in Fig. 5: the most probable values of x1 increase steadily with time, while those of x2 remain fixed.
In this situation, the ball’s behavior has nothing to do with the projectile. We cannot learn anything one way or the other about the projectile from the position or motion of the ball.
What about when a measurement takes place, as in Fig. 4? Again, as seen in Fig. 7, the majority of the wave function follows the path of the star, with the most probable values of x2 beginning to increase around the most likely time of the collision. This change in the most likely value of x2 is an indication of the presence of the projectile and its interaction with the ball. [Note: Fig. 7, unlike other quantum wave functions shown in this series, is a sketch, not a precise solution to the full quantum equations; I simply haven’t yet found a set-up where the equations can be solved numerically with enough precision and speed to get a nice-looking result. I expect I’ll eventually find an example, but it might take some time.]
More precisely, because of the collision, the motion of the ball is now correlated with that of the projectile — their motions are logically and physically related. That by itself is not unusual; all interactions between objects lead to some level of correlation between them. But this correlation is stable; as a result of the collision, the ball is highly unlikely to be found back in its initial position. And so, when we later look at the trap and find it empty, this does indeed give us reliable information about the projectile, namely that at some point it passed through the trap. (This type of correlation, both within and beyond the measurement context, will be a major topic in the future.)
So far, this all looks quite straightforward. The motion of the star in Fig. 4 is seen in the motion of the peak of the wave function in Fig. 7. Similar behavior is seen in Figs. 5 and 6. But these are simple cases: where the projectile’s motion is well-known, its location is not too uncertain, and the measurement device is almost perfect. We will soon explore far more complex and interesting quantum examples, using this simple one as our conceptual foundation, and things won’t be so straightforward anymore.
I’ll stop here for today. Please let me know in the comments if there are aspects of this story that you find confusing; we need all to be on the same page before we advance into the more subtle elements of our quantum world.
8 Responses
I have a question: the wave function in the measurement example gives the probability (density) of finding the ball and the projectile at the given locations when measured. Do the two occurrences of the word “measure” in the preceding sentence refer to the same thing?
OH! Good spotting. No, these mean different things. That’s very confusing and I will fix it.
You’ve put your finger on a tricky point that I’m going to have to figure out how to discuss more precisely. Even the language I’m using is inherently confusing and I’ll need to refine it. Thanks for the question.
The issue is that the ball, as a measurement device, does some measuring, but then we, in turn, have to measure the measurement device with yet another device, and also measure the projectile with yet a third device, if we want to actually do something that is related to|Psi(x1,x2)|^2 at a particular time. The fact that measurements are often chained together (consider the photograph of the meteor crater, for instance) is one of those subtleties that I was trying to avoid in this first post on the subject, but I guess I didn’t really manage to avoid it after all…!
Typically what we would actually do is we would let the first measurement occur, then we would later go look for the ball to see what the result of the original measurement was, while also verifying the original measurement by looking for the projectile. Whatever methods we use, and whatever x1 and x2 we find at that later time, they would tell us that |Psi(x1,x2)|^2 was at that time far from zero. [We wouldn’t really check that |Psi(x1,x2)|^2 gives us a probability distribution unless we repeated the experiment many times… which is sort of cheating, if you think about it… more on that another time.]
Thank you for this. I know there’s further to go down this road, but your definition of what constitutes a measurement has already given me a dramatically better intuition. The common confusion that arises from the Copenhagen interpretation ascribing a special role for consciousness couldn’t even get started if this definition of measurement was broadly accepted.
I’ve not come across this (or anything like it) before. I’ve seen plenty of people saying words to the effect of “measurement ≅ decoherence” (which I assume is roughly where you’re headed?), but without laying out the groundwork like you’ve done here, it’s very difficult to really understand what that means.
Glad to hear it. Yes, we’ll be able to see a simple model of decoherence that should clarify how it works. It will take us a while to get there because I want to be really thorough.
Another (historically relevant) quantum measurement is a cloud chamber.
Definitely, but much more complicated than an even simpler and equally relevant one: a Geiger counter.
Dr.Strassler:
I enjoyed that article very much. I have often thought about what constitutes a “measurement” I had always felt that some exchange of momentum or energy was enough to constitute a measurement.
Regardless if it is registered by a human or not.
For instance, in order for me to see something, photons must leave that object, either by bouncing off the object, from some eternal light source, or by being omitted from the depths of the object itself (black body radiation) when the photon leaves the object, either by bouncing or coming from its internal structure, the photon carries momentum & energy, and whatever it hits, my eyeball for instance, will record it. Even if I can’t “see” it optically, don’t infrared & ultraviolet photons register as energy, in my eyeball? I’m definitely not saying a human interaction is required, just that the firing off of a photon, regardless if it hits anything or not, is enough.
In an atom, if an electron drops down from an excited state, it omits a photon. Now, the when & where the electron deexcites is random, but when the electron deexcites, does not the atom recoil from the momentum? Wouldn’t this recoil be a “Measurment” of this interaction/exchange?
For me, the fact that the firing off of the photon is random, is not as important, as the fact that the “record” of this event is recorded in the recoil.
No, those are not necessarily measurements. These are physical processes, sure, but not all processes are measurements. Photon emission is not necessarily a measurement.
For instance, if you take the point of view that every photon emitted from every atom is a measurement, you have the problem that most of those photons are then absorbed by other atoms within the same material, on a time scale of a nanosecond or less. Worse, the photons are emitted in quantum states where even the existence, timing and direction of the photon is not known, and they interfere with others, and so you can’t necessarily work backwards to interpret the observation of a photon in terms of what it was emitted from.
A measurement has to lack ambiguity, whereas most physical processes in quantum physics allow for a great deal of ambiguity. I’ll try to find ways to be more precise about this in future.