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.
32 Responses
:…the problem of measurement: cutting to the chase and your interest in higher dimensions relative space time the space of possibilities set by the Higgs pertains over Vacua Thea or many potential vacuum levels as per string theory… does this mean the integrating process used per Hodge numbers in Calubi-Yao smooth Kahler manifold over split in holomorphic groups might be thought of as a kind of fuzzy number dynamic indices which might pertain to the expansion of the universe as being entertained over just such variation?
To simulate the interacting quantum case, have you though of using a simple Bethe ansatz solution that would describe to bosons with a delta potential of strength g? The two-body wavefunction is then just a sum over plane-waves, with coefficients that depend on the interaction. It should not be difficult to then create the two wave-packets.
I haven’t, no. That’s a good suggestion, thought it has been a long time since I played with the Bethe ansatz and so I have forgotten what cases it covers.
Some current questions on measurement where: the wave function become material seems to have overtones of palindrome numbers and “double magic” shells in physics such as lead thus the tension between xrays and lead for example… strangely, the three dimensional sphere is considered dimensionally a two dimensional n- phenomenon because it structure the surface the same… in your illustrations the Schrodinger structure seems to enter the third dimension via a time shift in the physical configuration which is like introducing an affine point structure in projection from surf ac of globe.. the self mapping of a circle, as a singleton become winding number does it seem to me to maintain over the wavicle view some reference to the point in a transform to a saddleback point?… meaning the self mapping itself which pertains to the d brand for example in precisely the opposition of poles?
There seems to be an exactness problem regarding the light from distant Quasars or other solar bodies. Everything is supposedly equated from the Big Bang but not everyone is on board with the Big Bang. So who’s measurement is correct under the current ‘theoretical environment? We have a problem with the concept that some ‘unmeasured’ comet slammed into earth missing all the Birds, and somehow the atmosphere was still conducive for breathing for these birds, whereby they ‘just flew up’ and they all survived. A bit like someone flew over the cuckoos nest. However, It is agreed that ‘measurement’ is a stable qualifier in all worlds, cosmos. universes, otherwise life would be pretty unstable like Randy McMurphy. Science and Theology in its absolut truths, solves many of lifes problems and answers untold questions Without measurement all the senior citizens would have issues with medicines and there would be countless deaths and doctors would be mass murderers.
So thanks to Science that we are in a stable society (excepting those in the White House), As for us we are happy under our umbrella of theology that is not in competition with Science but live under the same roof. Again thanking you for your research and making things known
Cosmologists, the experts here, are onboard with space expansion (the most general definition of “Big Bang”). [C.f. the many papers on the topic.]
Hi Matt. If I have understood correctly, in 1920s QM an electron has a definite momentum and a definite position, it is just that we can’t know both simultaneously. Just a minor point – as it has a definite position, does 1920s QM assume that an electron is a point particle – if not it seems that it would need to add a further definition saying something like the position of the electron is the position of say is centre of mass?
The main point of my question is to understand what it is about experiments that precisely measure the momentum of an electron that means we lose all information about where the electron is. It would be great to know exactly how a precise measurement of momentum is made, so that we can understand why this leads to us knowing nothing about its position. Or is 1920s QM saying that when we exactly measure the momentum of an electron, this causes the electron to be infinitely spread out, and a subsequent measurement of its position would cause it to shrink to a point, but that measurement can be performed at any point in space because it is infinitely spread out, so the position of the point electron discovered by this subsequent measurement could be at any point in space?
QM does treat an electron as a “point” particle to within its limitations. It does so in the following sense: it associates to the electron a position [and momentum], a spin of fixed magnitude, a fixed electric charge, and nothing else. A non-point particle would have additional degrees of freedom; for example, atoms and molecules have multiple quantum numbers, which can be very large, and that distinguish large numbers of states of the atoms or molecules with different energies.
But QM cannot both do this correctly and be consistent with special relativity, which is why QFT is neccessary.
As for momentum measurements: your last sentence is correct, although the logic language is a little off.
In QM language: If we have a beam that sends out identical electrons once every second at high speed, for which the relevant part of the wave function is initially spread out completely in a simple sine wave that is multiple meters long, then different measurements of each electron’s momentum will give almost exactly the same answer each time.
If the electrons are less spread out, then the answers will be somewhat more variable.
The question of what the relevant part of the wave function will look like after each measurement is more complicated. There is no guarantee that the electron’s momentum following the measurement will be what it was before the measurement. Non-destructive measurements which leave the momentum with the exact momentum that it had during the measurement are not realistic, and would require an elaborate device; there must be limitations. For instance it may take an infinite amount of time to do such a measurement, which would result in an electron whose position is completely unknown. I will have to think about how to design such a measurement. In reality no one ever measures a momentum to arbitrary precision, nor does anyone expect the electron to be described with a wave function after the measurement that has zero uncertainty, so the spread of the electron is never infinite. [Measurements of discrete quantum numbers, such as whether the spin points up or down relative to a magnetic field, are much easier to discuss, because there are only two outcomes, rather than a continuum of possible outcomes with a spread that no realistic experiment can completely eliminate.]
The language in quantum field theory is different because the electron itself is spread out, not the wave function of the electron field. But we’ll have to get back to those sorts of questions at a later time.
As for how to do a measurement of momentum that is as exact as possible; let me think about this further, as it is a good question deserving of a careful and precise answer, which I don’t think I can give yet.
Much appreciated Matt. I understand how the uncertainty principle between position and momentum arises from the mathematics of QM, but I think it would be very insightful to also understand how it arises experimentally. I have never seen that discussed.
Hi Matt. The more I think about the QM uncertainty principle for position and momentum the more problematic it becomes I am afraid. Even if we forget about exactly how the momentum of an electron is measured, the uncertainty principle is saying that if we precisely know the momentum of an electron on Earth, then it is equally likely to be found anywhere in the universe. But that would contradict the speed of light as a cosmic speed limit, as if we measured the position of the electron say a second later, then the electron could not possibly be more than 300,000 km away from Earth. But the uncertainty principle is saying that it is just as likely to found a million light years away as it is to be found on Earth? Even if we say that it is impossible to measure the momentum of the electron exactly, surely it is possible to measure it sufficiently precisely that the electron has a non-zero probability of being a distance away from Earth that would require travel at faster than the speed of light?
You’re not wrong to think these things through carefully, but you are in some danger of confusing some issues here.
Nothing in the uncertainty principle tells you how to produce the state you want. In particular, it doesn’t tell you how long it might take. Yes, relativity tells you that it is very difficult to produce a state with exactly known momentum. However, it is easy to see how to make a state with arbitrarily well known momentum, if you’re willing to wait.
Make a box in the shape of a cube of size L; maybe it’s a centimeter, maybe it’s a meter, maybe a kilometer. Put it within another metal box to isolate it from the cosmic microwave background, and cool it to as close as you can get absolute zero temperature. Put a single electron in it. And WAIT.
The electron will rattle around in the box, but very gradually it will radiate its excess energy away in low-energy photons until — after a long time, and if the box is cold enough — it is in the ground state within the box. The part of the wave function that involves the electron will have a constant wave function across the box, and momentum as close to zero as it can be within a box of length L.
Now be quantitative: if the box is 1 meter across, the uncertainty in its position in the ground state is L in each direction. So the uncertainty in the electron’s momentum, divided by its mass, gives us a speed of no more than 0.01 millimeters per second = 10 microns per second = 10,000 atom-lengths per second — assuming you can get the temperature down to 2 milliKelvin. If the box is 1 km apart, then you’re down to 10 atom lengths per second, and require the box be at a temperature at 2 nanoKelvin.
Right now I don’t know a super-quick way of calculating how long you’d have to wait, but I’m sure I could figure it out if I gave myself ten minutes to think about it; you just need to estimate radiation rates, and there must be an easy way using dimensional analysis.
But of course this is impractical. The universe is filled with cosmic microwaves at 2 Kelvin, and any large terrestrial lab is at higher temperature than that, and no one is going to clear out and cool a space 1 km on a side. So in practice this cannot be done. But you see, this is a practical problem, not a problem of principle. And there is no problem with relativity, since the universe will be around for a long time and I can take as long as I want to set up this state.
So in conclusion, what you have learned is that you cannot set up a state of constant momentum, or hope to measure it, without requiring a very large amount of time. And that is no surprise, because having constant momentum is related to having constant energy, and in relativistic theories there is a sort of similar relation Delta E Delta t ~ h/4 pi, which says that to measure energy precisely requires infinite time. All consistent with special relativity.
Hi Matt. Just want to make sure I have got the physics right. To calculated the speed of the electron
uncertainty in momentum approx equals hbar/2 divided by L
speed of electron approx equals uncertainty in momentum/mass of electron
Temperature is then calculated using
3/2 kT approx equals kinetic energy of electron, with kinetic energy calculated using 1/2 mass x velocity squared
Have I got that right? Only thing that worries me is 3/2 kT gives the average kinetic energy of a very large number of electrons at temperature T, but can’t really be applied to a single electron, which doesn’t really have a temperature.
The temperature I’m referring to is that of photons coming off the walls of the box; they will in turn influence the electron’s momentum through photon-electron collisions.
Hi Matt. Just trying to calculate this temperature. If we take an electron travelling at 0.01 mm per second, it has a kinetic energy of of 4.5 x 10(-41) J, where I am using the bracket to indicate a power. If we use Wien’s displacement law, at a temperature of 2 milliKelvin for the walls of the box, the most common photon emitted will have an energy of 7.8 x 10(-26) J. Surely this is much too high and we need a much lower temperature? How have you calculated the temperature?
It’s certainly possible I messed this up by neglecting a square root somewhere. I’ll try to do it again, and if there’s an error I’ll report it here and fix it in the original comment. The general point doesn’t change of course.
Einstein included the cosmological constant as a term in his field equations for general relativity because he was dissatisfied that otherwise his equations did not allow, apparently, for a static universe : gravity would cause a universe that was initially at dynamic equilibrium to contract. To counteract this possibility, Einstein added the cosmological constant. However, soon after Einstein developed his static theory, observations (the development of the Standard Model was driven by theoretical and experimental particle physicists alike. The Standard Model is a paradigm of a quantum field theory for theorists, exhibiting a wide range of phenomena, including spontaneous symmetry breaking, anomalies, and non-perturbative behavior or X1, X2 = 0) by Edwin Hubble indicated that the universe appears to be expanding; this was consistent with a cosmological solution to the original general relativity equations that had been found by the mathematician Friedmann, working on the Einstein equations of general relativity.
In the Special Theory of Relativity, Einstein determined that time is relative—in other words, the rate at which time passes depends on your frame of reference.
If there were no observer, spacetime would still have its “time” dimension. It follows that it is false that for general relativity time is relative (because it is a dimension of spacetime, which is not relative).
So between the cold-spot in the Cosmic microwave background & the observer, there is time dilation appears as ‘space expansion’, but because of Diffeomorphism or gauge invariance (the parameter that you change to go from one configuration or hidden variables to another with the same state is called a parameter with gauge invariance), the “physical reality of Time dilation & Rest mass Unnaturally coexists” or the “the equivalence principle” holds.?
Having participated in a thought experiment with Prof Asher Peres, on how often are Bell’s Inequalities violated? I know about a set of measurements. I told my granddaughter who is very smart about a Stoppard play in which there were 85 heads, on tossing a coin. I said “impossible”, she said “improbable”. Faced with the
“impossibility” of teacher about probability and measurements, I evaded the question with a small “lie”. I said I didn’t remember the math. But that the coin “remembers” the previous throws.
susanjfeingold.com
Thanks very much for this series of articles. It has clarified some things about quantum mechanics for me that a lot of pop-science treatments leave hazy (in particular your emphasis on the difference between physical space and the configuration space of a system).
I wonder if there is anything to be said about the little ripples that seem to appear on the main “hump” in figure 7 after the collision. Are those real or artifacts of the simulation process? If they’re real, what information about the collision are they reflecting? It’s hard to see them in enough detail to know what they’re actually doing (e.g. what angle they’re at relative to the coordinate axes).
Figure 7 isn’t calculated carefully; it’s a sketch. I’m working on getting some cases calculated where the details are right, which requires finding a case that isn’t swamped by numerical artifacts. But in any case, the details will depend on exactly how the projectile interacts with the ball, and I haven’t specified that here. In other words, we can get a lot of different behaviors depending on the interaction we assume between the projectile and ball, and only the qualitative features shown in the sketch will always be true.
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 much later time.
[We wouldn’t really be able to check that |Psi(x1,x2)|^2 actually 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.
Ahh ok, I understand, I was looking at it from the standpoint of the example in the article, of the projectile colliding with the ball. During the collision, the projectile transfers momentum & energy to the ball, and as such the displacement of the ball is a “measurement” of the projectile passing. Where as, when the projectile misses the ball, no momentum & energy is transferred, and the ball does not register the passing of the projectile.
Indeed, what makes this a measurement is that in this case the ball’s shifted position is a nearly unambiguous measure of the projectile’s presence. If the ball were being pounded by projectiles from all sides, we could not conclude this; or if the displaced ball were immediately subject to a force back to the left that pushed it back into the trap, so that the transferred momentum was quickly lost to some other objects, again we could not draw a conclusion. It’s really important, in Fig. 2, that the ball is initially stable and that the departure of the ball from the dip to the right is not easily reversed.
Dr.Strassler:
I think I probably misunderstood what Dr.Susskind said during one of his presentations, that if one atom records a passage of event, that counts as a measurement. Again, I’m sure I misunderstood his comment.
I’m sure he was implicitly saying what I’m saying here: the crucial word is “records”. Not all processes leave an unamibiguous record, even for a brief moment. Depending on the circumstances, ionizing an atom, or destroying its nucleus, or giving it an unusually large speed can create an irreversible effect (or an effect which takes a long time, at least on atomic scales, to reverse). But suppose the atom is in a hot gas that has it colliding with other atoms and moving randomly at hundreds of miles per second; if it is then impacted by a particle that changes the atom’s velocity by just a few feet per second, that impact is not going to leave a record.
Actually my answer here is imperfect, and I’ll be refining it next week or the following, by looking at the issue of what leads to something being “recorded”. But we need to build up more intuition first for simple situations before getting into issues like decoherence.