What is the uncertainty principle?

Question

I looked on Wikpedia for information on the uncertainty principle, but after reading it I still had no idea. I know it has something to do with how many things you can hold at some spot for some amount of time (maybe?). This is inspired by this question.

Answer 1

Let's forget physics for a moment and just talk about the mathematics of waves.

The uncertainty principle is a property of waves. Think of a single, narrow pulse traveling along one direction. The pulse is narrow, and so the position of the pulse at any given time is easy to quantify. But this is a single, non-periodic pulse. You can build up such a pulse from a lot of sinusoids, but in doing so, the wavenumber (frequency) of such a pulse becomes ill-defined. It's a combination of a lot of wavenumbers, so no single wavenumber describes it.

Conversely, think of a pure sinusoidal wave. It has a single wavenumber, but what is the position of such a wave? It a periodic thing; it doesn't have a single position. You can describe it in terms of magnitudes at a lot of positions, but that's all.

That's the uncertainty principle, a fundamental property of waves. You can localize a wave in terms of its position or wavenumber, but not both at the same time (at least, not below a certain limit). The quantities $\sigma_x$ and $\sigma_k$ describe the distribution of positions and wavenumbers for a given wave (smaller $\sigma_x$ means that the wave is more like a localized spike at one position; larger $\sigma_x$ means that it's smeared out more like a sinusoid), and their product is constrained by the uncertainty principle.

The uncertainty principle is usually phrased in terms of position and momentum, instead of wavenumber as I've written. This is just a little bit of physics: we say that momentum is proportional to wavenumber by the constant $\hbar$.

Answer 2

In classical physics you are supposed to be able to measure the coordinates and the velocity (really the momentum) of a mass with infinite precision at the same time. If you try this trick in the lab you notice that that's not the case. Either your position or your momentum measurement or both will always show some non-trivial statistical fluctuations when you repeat your experiment many times. If you multiply the standard deviations of these fluctuations with each other, no experiment that you can ever perform yields a product that is smaller than a certain number. That is it.

Answer 3

There is a related (and relevant) question with answer called Uncertainty principle and measurement but I want to give a specific answer for this situation and background. You will need top know the difference between precision and accuracy. A precise dart thrower throws darts that land very close to each other, this isn't a deep statement, in fact it is just a definition of the word precision as it is used in science. An accurate dart thrower throws darts that average out near the center, this also isn't a deep statement, in fact it is just a definition of the word accurate as it is used in science. A precise dart player might throw them all at the same small upper region just barely on the dart board. If only they could aim more accurately they would be awesome. Maybe they are blind and someone is describing the center of the dart board poorly, but they are precise, they can control their variability very very well. An accurate dart player might throw them so they always hit the middle half of the target, distributed equally in that region. They know where the center is very accurately a better description wouldn't help, they have totally accurate image of their target, they just can't reliably throw the same way each time, they have control over not being too high or low in general but can't control how off they are each time.

The same things happen for experiments.

In general, the uncertainty principle relates how a particular experimental setup sometimes has a tradeoff between how variable the results of different measurements could turn out. The precision of one measurement comes at the expense of the precision of other measurements. And I hate the name measurement because it makes that sound totally crazy. Just think of it as the variability of the the results of one interaction coming at the expense of the variability of the results of other interactions.

Imagine you setup your equipment but you don't know who is going to come in or what they are going the measure. In Newtonian physics you can just set it up precisely and whatever they measure they will get results based on how precisely you set it up. (Precision being how reliably you set it up, accuracy being how close to what you aimed for, but this discussion is all about precision (reliably consistent, not variable), not accuracy). In quantum theory there is a tradeoff, no matter how you set up your system some people will find it a more reliably setup system then other people will find it. And whether they find it reliable will depend on what they choose to measure. So we need to discuss variability so that we can discuss and quantify how well it can be done.

One way to measure the variability of a measurement $A$ is to first look at the average of many results, denoted $\langle A\rangle$. Sometimes people put a line over it as an alternative notion, and sometimes people call it a mean rather than an average. OK. But a mean or average doesn't tell you how variable is, it really tells you more about the accuracy, whether you are on average over doing it or under doing it. But if you always got the same result, then your results would always be that average result, so you can look at the result of measurement $A$ and see if it is above or below $\langle A\rangle$, which is the same as checking whether $A-\langle A\rangle$ is positive or negative or zero. So for a low variability result (a precise result) that number will be close to zero. However it will on average be zero, regardless of how precise or variable you are. So we can't just discuss the average of $A-\langle A\rangle$ since $\langle A-\langle A\rangle\rangle=0$.

But we can look at$ \langle \left( A-\langle A\rangle\right)^2\rangle$ and since each result of $\left( A-\langle A\rangle\right)^2$ is zero or positive, we don't expect the average to be zero unless it is zero with probability 100%.

So that's a way to measure variability. We said there was a tradeoff. The tradeoff is that sometimes a decreased variability for one measurement comes at the expense of an increased variability of other measurements.

I'll get into that later. But first I want to mention that when people say "the" uncertainty principle they often mean the particular tradeoff between position measurements and momentum measurements. So they will say the variability of position and the variability of momentum have a tradeoff in that they can't both be small.

But we do need to be clear about what the tradeoff is.

Step 1, select a state $\Psi$. You can select any state you want.

Step 2, prepare many systems in same state $\Psi$ (you need many to see how reliable/precise or variable you are).

Step 3, select two operators A and B (these correspond to the interaction/measurement, as mentioned people probably assume position and momentum as the two operators/interactions/measurements)

Step 4a, for some of the systems prepared in state $\Psi$, measure A (mathematically you use the operator to make the prediction, in reality you do the interaction, and linguistically you call it a measurement even though that is a horrific name)

Step 4b, for some of the systems prepared in state $\Psi$, measure B (same deal, different operator/measurement/interaction)

Now you analyze the results. Every time you measured A, you got a a result. (Mathematically every time the oeprator A came up you got an eigenvalue of A), and similarly for B. Each result had a probability (which is equal to the ratio of the squared norm of the projection onto the eigenspace divided by the squared norm before you projected onto the eigenspace, but that doesn't matter here). So your results of A come from a probability distribution that often has a mean $\langle A\rangle=\langle \Psi|A|\Psi\rangle $ and a standard deviation (that measures the variability) of $\Delta A=\sqrt{\langle \Psi|\left(A^2-\langle \Psi|A|\Psi\rangle^2\right)|\Psi\rangle}$. And your results for B come from a probability distribution that often has a mean $\langle B\rangle=\langle \Psi|B|\Psi\rangle $ and a standard deviation $\Delta B=\sqrt{\langle \Psi|\left(B^2-\langle \Psi|B|\Psi\rangle^2\right)|\Psi\rangle}$. You don't get those from a single measurement, o they are averages. You don't even get them from a whole bunch, but from steps 4a and 4b you do get a sample mean and a sample standard deviation (which are mean and deviation of the actually experiments you did not the mean of the unlimited number you could have done but didn't), and for a large sample these are likely to be very close to the theoretical mean and the theoretical standard deviation.

The uncertainty principle says that way back in step 1 (when you selected $\Psi$) you could select a $\Psi$ that gives a small $\Delta A$, or a $\Psi$ that gives a small $\Delta B$ (in fact if $\Psi$ is an eigenstate of A then $\Delta A=0$, same for $B$). However, $$\Delta A \Delta B \geq \left|\frac{\langle AB-BA\rangle}{2i}\right|=\left|\frac{\langle\Psi| AB-BA |\Psi\rangle}{2i}\right|,$$

So it's saying that the product of the two measures of variability $\Delta A$ and $\Delta B$ each of which is not negative have to have a product that is at least $\left|\frac{\langle\Psi| AB-BA |\Psi\rangle}{2i}\right|$ (itself a kind of average). When A and B are position and momentum, that thing is $\hbar/2$, so if $\Delta B < \hbar/2$ (numerically) then $\Delta A > 1$ (numerically).

So this explains why people bring it up. They say that when you claim to balance the pencil it sounds like you have a never small $\Delta X$ where $X$ is position, and that therefore $\Delta P$ must be large (where $P$ is momentum), so they want to say that it must have a large variability in momentum, hence velocity, hence can't be balanced.

But that is misleading. The uncertainty principle talks about the distribution of results for many systems prepared the same way under particular interactions. And the inherent tradeoff that when results of one interaction (say momentum) are low variability then the other one (say position) his highly variable.

The real problem is that things like electric fields are quantum too, so matter what state you put them in, there is some uncertainty, so they are never truly "not there". You can try to have as small an electric field as possible but at some point there is a tradeoff even for the state we call the vacuum. So photons (light, electric fields) can interact with your pencil. And your pencil is capable of falling in a way that gives energy to the photons, and you can't really stop that from happening. The quantum photons can break the symmetry of the pencil even if you try to minimize their effect.

Unfortunately the uncertainty principle for the photons is not the momentum-position uncertainty principle that people usually bring up. Which is why I brought up a more general one. Some people will call the minimal electric field a zero point field or discuss a zero point energy. The point is that it can steal energy from your pencil and you can't stop it. And since the electric field can come from elsewhere and can't go away entirely no matter what you do, it is unavoidable.

Answer 4

Let $X$ and $Y$ be random variables with density functions $f$ and $\hat{f}$, where $\hat{f}$ is the Fourier transform of $f$. Then the uncertainty principle is a lower bound on $\sigma_X\cdot \sigma_Y$ (where $\sigma$ is the standard deviation). In particular, $\sigma_X\sigma_Y\ge 1/4\pi$.

For example, take a particle moving in one dimension, in a fixed quantum state. Let $X$ be the observed value of the particle's location (if you choose to measure its location) and let $Y$ be the observed value of its momentum (if you choose to measure its momentum). Then $X$ and $Y$ satisfy the assumptions above, and hence satisfy the conclusion.

Answer 5

A highly simplified analogy that attempts to get thinking going in a direction that can sometimes lead to eventual understanding...

Think of various photographs taken of a baseball in flight. Different photographs are taken with different exposures. They are part of an attempt to measure both speed and position at the same time. In particular, imagine that one photograph is "instantaneous".

All of the photographs except the "instantaneous" one will show the baseball as something of a smear because the exposures caught the ball in motion. Longer exposures will show longer smears.

By knowing the exposure times, you can measure the lengths of the smears and determine how fast the baseball was moving. The longer the exposure, the closer you can get to determining the speed because the length of the smear can be measured more precisely. If measuring the smear has a measurement error of half a millimeter, then a 10 centimeter smear gives a more precise result than a smear that's only one millimeter long. Unfortunately, the longer the smear, the harder it is to say exactly what the "position" is because the position was continually changing.

The smear helps some with speed, but it still doesn't tie down all related details, e.g., it doesn't expose whether travel is from left to right or right to left. There is much more to the problem for precision on the much smaller scales where the uncertainly principle becomes more important.

Then think of the "instantaneous" photo. In that exposure, there was zero smear. It's a perfect image of the ball at one/single point in its flight. But now, by looking at your 'measurement', what can you tell about its speed? Or even about its direction? The ball would look exactly the same if it was going left to right across the photo or falling straight down or if it had been tossed up in the air and had just reached the highest point and was reversing its direction. With a perfect measurement of position, you totally lost all information about motion.

Now, a physical baseball is not a great expression of the details about particle physics. Unfortunately, nothing we experience in 'the real world' is. But if we want simple analogies, we're stuck with incorrect examples. Still, by using real-world examples, it sometimes becomes possible to use the terms to illustrate the bizarre happenings in quantum realities. Maybe it helps lead to understanding for you, maybe not. Just always be aware that analogies are not the realities and and are never correct.

Answer 6

user2338816 put forth a photography analogy. I like photography analogies, but I have a different one to share which I believe is a hair closer to the full story.

First off, to answer the question directly, the uncertainty principle is that the product of the minimum error possible in your measurements often limited. The most famous example is position and velocity. If you measure position and velocity of a particle (QM scale), you will have some errors. Some of it is because your machine is imprecise, but the Uncertainty Principle states that no matter how good your measuring equipment is, you will never get that product below h-bar/2 (ΔxΔp >= ℏ/2).

Why do they claim this? Well the really short answer is because the math of QM says it should be so, and nobody has ever found an experiment that contradicts the claim. The long version is that quantum mechanics claims that everything is a waveform, and there methods of generating a "classical measurement," like position or velocity cannot capture "all of the information."

Here's my photography metaphor. We're going to take a picture of a string on a violin after it has been plucked. We know from the mathematics of how strings vibrate that there should be a standing wave. We should see one of those nice sinusoid waves we see in mathematics class.

However, our camera has limits. It has to see photons to resolve an image. Arguably it measures the photons which interact with the string and find their way to the lens. To get enough photons, we keep the shutter open a long time. This means, instead of a nice clear sinewave, we get a blur whose envelope is definitely the magnitude of the standing wave. We can even calculate the velocity of the particles going up and down on that string, we just don't know their position in the up/down axis.

So lets use a strobe. Strobes emit a huge number of photons rapidly, so that they interact with the string in "almost an instant." With this, we can collect a bunch of photons that show the string, clear as day. Now we can easily see exactly where the string is, but we've lost velocity information. The string looks like it is holding still.

Now we're clever people, so we can make a strobe that fires over a larger time period, so we can get "decent position information" and "decent velocity information" from a picture where the string has some blur (from a long strobe time), but much less blur than the first picture, so you can get some position information as well.

If you squint your eyes at this analogy, you can see the first layer of the uncertainty principle forming. There's a tradeoff between how well you can measure the velocity of any part of the wire against how well you can measure its position. You're free to play the tradeoff game, but you can't exceed the physical limits of the system.

So now we solve the problem by taking two pictures. In the classical world, this is a totally valid technique to measure both position and velocity -- we do it in highschool physics classes. Two super-bright strobe flashes later, we have two super-crisp pictures of the string in different positions. We can do some math between the pictures, calculate the velocity and position of the wave at both times, and then we can predict the state of the wave forever. Easy! And very classical.

In the quantum world, the rabbit hole goes deeper. As you get smaller, a discrepancy between classical and QM measurements comes to light: your machine is no longer a passive observer -- it is part of the system. The flash bulb had always affected the string by bombarding it with photons. However, at the macroscopic level, this didn't do much to the string at all. It behaved just as though you were not there.

However, get smaller, and your measurement apparatus starts to interact with your subject. For the photography analogy, the photons from the strobe start to shove the string around in highly unpredictable ways (the actual QM situation is more complicated, but I find this analogy is sufficient for understanding). The strobe flash of the first picture dramatically changes the velocity of the string, as the photons bombard it (interact with it, in QM speak). The same goes for the second one.

When we try to run the math in this case, we find that the strobe flash effectively invalidates a great deal of the information we thought we knew about the system before hand because it perturbs the subject. The mere fact that we used a strobe flash strong enough to apparently freeze the string in space cause it to disrupt the velocity of the particles so greatly that we lost nearly all information about it.

And so, the uncertainty principle is the real life version of this photographic metaphor. It is "true" simply because we've measured this behavior, and never once have we found a way to get around it, no matter how hard we try.

Is it possible to measure a quantum state exactly, using classical terms? Maybe. The universe is a strange place, and we're only just starting to understand it. However, to date, not only has there been no theory showing how to do it, most theories proposed explicitly forbid it. It's simply the world, as best as we know it!