How can I accurately state the uncertainty principle?

Question

In almost every introductory course, it is taught that the uncertainty principle happens due to disturbance in the system to be measured. Teachers give these examples that induce students to misunderstand the phenomenon.

This has probably been going on for a long time. Heisenberg used to illustrate his uncertainty principle by giving the example of "taking a photograph of the electron".

To take the picture, a scientist might bounce a light particle off the electron's surface. That would reveal its position, but it would also impart energy to the electron, causing it to move. Learning about the electron's position would create uncertainty in its velocity; and the act of measurement would produce the uncertainty needed to satisfy the principle.

However, the uncertainty is not in the eye of the beholder (see here).

Many students, like me, continue their studies of quantum mechanics while ignoring the conceptual subtleties of the uncertainty principle, and this generates a snowball effect. These students do not have solid knowledge of the subject, which is already intrinsically not very intuitive. Later, they learn that there is an uncertainty relation whenever two operators are non-commuting. However, this is a rather abstract definition.

I'd like to know if there is a simple conceptual explanation that uses key concepts such as "observation", "measurement", "decoherence" accurately. For example, if I were a teacher, how could I state Heisenberg's uncertainty principle in a less mathematical, but conceptually rigorous way?

Answer 1

As pointed out in a comment by mmesser314, there is a video by the mathematician Grant Sanderson.

The more general uncertainty principle, beyond quantum

Grant Sanderson puts the HUP in the context of properties of wave propagation.

In the case of a continuous, consistent oscillation the corresponding propagating wave has a specific frequency, but that propagating wave doesn't have a location; it's continuous.

It is possible to produce a burst of oscillation in such a way that it gives rise to a propagating "blip". A fourier analysis of the waveform of that blip describes it as a superposition of a range of frequencies. In the case of that blip: the location of it can be tracked through time with specifity. But the blip does not have a particular frequency; the blip is spread out in frequency space.

Grant Sanderson describes that with wave propagation in general (not just in the context of quantum mechanics) there is an inherent trade-off. You can push for a very specific frequency, but at the cost of specifity of position-as-a-function-of-time. You can push for high specifity of position-as-a-function-of-time, but at the cost of introducing spread of spectrum.

In any device that produces propagating waves the design can be made so that the emitted wave is wherever you want in that trade-off.

Fourier analysis facilitates expressing the trade-off in mathematical form.

Stating it explicitly:
The view in terms of wave propagation in general does not need to invoke concepts such as "observation", "measurement", "decoherence".

Answer 2

Short version: "There are pairs of variables in our universe that cannot be simultaneously measured to more than a certain amount of accuracy. One such pairing is momentum and position."

That's where I'd start for students new to the concept. Possible additions to that version:

• Other pairs include angular position with angular momentum, or energy with time.
• This uncertainty is always there when you measure those pairs of variables. It's not a problem with how we measure things, and it won't go away with better measuring tools. It's an inherent part of our universe.
• If you construct a clever situation with a very high amount of accuracy in one variable (like by passing a beam of particles through a small space so their position is constrained), the paired variable's uncertainty will be very high (the particles will spread out more after they pass through th space).
• You can measure unpaired variables at the same time just fine. If you measure linear position and angular momentum, for example, you can get as precise as you want.
• There are mathematical methods that can describe this: there are simple less-than-or-equal-to equations, there are methods that use vectors and matrices, and there are methods that use wave equations.
• The amount of inaccuracy is pretty small - you won't notice in your daily life. This isn't for kicking soccer balls through a goal, it's for shooting laser beams through a barely-visible hole. But like we saw with the photoelectric effect, quantum things can still affect your daily experience.

I taught a little bit of quantum to my high school students, and this level of discussion was about right for them. Then we'd get into using $\Delta x \Delta p_x \geq \hbar/2$ some, because they had enough algebra to do that.

Answer 3

I have the impression that people prefer to stick to old steps in the history of Physics more than to use the facts as embodied in the full formalism of QM. Even worse, recent progress on this subject seems to be confined to a limited circle of specialists.

Let me try to state a few well-known and less well-known things about uncertainty relations (UR). It is not by chance that I prefer to call them UR and not Heisenberg Uncertainty Principle (HUP). Although Heisenberg should be credited for their first formulation, his interpretation and justification are not satisfactory for current standards.

I'll start with some sharp statements, and I'll discuss them.

1. in the modern view of QM, UR are not principles but can be derived as a theorem (Robertson's theorem). I.e., they are not a way of summarizing experimental facts but are a consequence of other basic assumptions of QM again based on experiments (observables represented by an algebra of non-commuting self-adjoined operators, probabilistic interpretation, the role of the eigenvectors, and eigenvalues, and so on).
2. the modern interpretation of UR is purely statistical and has no direct relation neither with practical limits of the accuracy of measurements nor with simultaneous measurements;
3. the problems of simultaneous measurements and the modification of the state induced by a measurement do not disappear. Still, they are separate from UR (although they share with UR a common origin).

About point 1. there is little to add. The derivation of UR from the definition of the uncertainty of two non-commuting operators is in every QM textbook. There could be some variation in the attention to the operators' domains, but the result should be crystal clear. The spread of the measured values of two non-commuting observables (like the x-components of position and momentum of one particle) must obey an inequality relation. How the underlying measurement has to be thought? We have to think of an ensemble of particles prepared in the same state. Then we perform measurements of position or momentum (only one of them!), and we collect the results. Each measurement (either position or momentum) is, in principle, performed with arbitrary precision; still, the results for position and momentum show a distribution of values. The two distributions of values depend on the nature of the common initial state and can be varied as a function of it. However, no change in the initial state can violate a relation between the two spreads of values as embodied in the UR.

I think that this way of presenting the interpretation of the UR is fully consistent with formalism and eliminates from the beginning the usual misconceptions about the precision of single measurements or the role of simultaneous measurements. That's the reason for my point 2.

What about point 3, which is related to the usual folklore about HUP?

Well, there is a trivial truth in the original Heisenberg's idea. Trivial, according to our ability to state a full list of postulates for QM, of course. Heisenberg was not in a similar position. The trivial truth is that if two observables do not commute, there is no possibility of finding simultaneous eigenvectors of both operators. The physical basis for that can be related to the modification of the initial state due to the measure, of course. However, it should be clear that a careful analysis of the consequences of the formalism on simultaneous or almost simultaneous measurements calls for an analysis completely different from that underlying the usual proof of UR. In particular, there is no reason to believe that the result should coincide with the form of UR.

Actually, in recent years this problem has been attacked from the experimental and from the theoretical side with interesting results (Ozawa, M. (2003). Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Physical Review A, 67(4), 042105, Rozema, L. A., Darabi, A., Mahler, D. H., Hayat, A., Soudagar, Y., & Steinberg, A. M. (2012). Violation of Heisenberg’s measurement-disturbance relationship by weak measurements. Physical review letters, 109(10), 100404, Busch, P., Lahti, P., & Werner, R. F. (2014). Colloquium: Quantum root-mean-square error and measurement uncertainty relations. Reviews of Modern Physics, 86(4), 1261. There has been some vigorous debate and I have the feeling that dust has not settled yet. However, I think that the bare existence of such recent debate is a clear proof that the usual way of interpreting UR as a statement on simultanous mesurements is an oversimplified and unjustified logical step. From pedagogical point of view I think this is a very good reason for presenting UR without paying a tribute to an old but uncontrolled tradition.

Answer 4

It would have to be this:

No agent can have more correct information about a pair of incompatible physical parameters than a certain fundamental limit.

A version that would perhaps imbue a bit more ontological interpretation, and so you could contest if you want, would be:

The universe contains a fundamental information limit regarding how much information it allocates amongst a set of non-redundant parameters one can define a physical object as having.

The Heisenberg principle is therefore best expressed in terms of informational entropies, which describe the relative degree of privation of information from an agent versus a system: the Shannon entropy, when an agent ascribes a probability distribution $f_X$ to an unknown quantity $X$, is

$$H_X := -\int_{x \in \Omega_x} f_X(x) \ln f_X(x)\ d\mu$$

and the uncertainty principle between quantities $X$ and $Y$ is

$$H_X + H_Y \ge K(f_X, f_Y)$$

in the most general. In the case of the positional and momental parameters on a single Galilean moving particle, we have

$$H_x + H_p \ge 1 + \ln(\pi \hbar)$$

which, as one may note, is going to be units-dependent: it is a general feature of the entropy of any continuous random variable that is always measured relative to a set reference level, and not absolute. The units on the right hand quantity are nats (natural unit of information), for bits, use

$$H_x + H_p \ge \lg(e\pi \hbar)$$

where $\lg$ is the binary logarithm. A citation for the above formulas is:

https://arxiv.org/abs/1511.04857

Not only is this perhaps more accurate in natural language, the mathematical statement is literally more accurate in that there are possible combinations of $f_x$ and $f_p$ which notionally satisfy the "deviation" version of the HUP but do not satisfy the entropic version, and yet also, cannot be validly combined into the positional and momental representations a quantum vector. (For example, consider a pair of delta spikes in both $f_x$ and $f_p$ at suitable separation. This may satisfy deviation-HUP, but it doesn't satisfy entropic-HUP.)

A crude model for a system that exhibits uncertainty-like behavior is the following. Suppose you wanted to make a computer game that had a ball in it, and you wanted to only use, say, 64 bits of information to encode both where it is and how it is moving. You could, say, use a 32-bit number for each, giving resolution to a level of $2^{-32}$, but if you wanted more for one or the other, e.g. a 48-bit number for the position (perhaps the balls are microscopic) then you must lose out on the velocity, e.g. only 16 bits, where we imagine describing both of these in the usual way as binary numbers along a coordinate axis. Note, of course, that either number must be relative to some base scale.

And the uncertainty principle basically says our real-life Universe works somewhat like this - but given the other details of the mathematics involved, the encoding and handling of that information can't be anywhere near as cheap and rude as simply truncating coordinate values(*). For one thing, theorems like Bell, PBR, etc. which seem to point to a certain holism as being involved. For another, though, no one single scheme is implied by quantum theory.

What that last part suggests is that quantum theory, in this view, is a subjective theory, but not absolutely such. It describes reality from our viewpoint within it, not from outside it, but that also doesn't mean that it has no relevance to "objective" physics at all. The uncertainty principle is one place where it connects with objective physics, for there are actual, physical consequences, in that we can observe them, of the act of extracting information from a physical system, and those consequences are governed in part by the Heisenberg rules. The constant $\hbar$, which sets where it happens, is the information content limit of our Universe, in the same way the constant $c$, the speed of light, is the information transmission speed limit.

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(*) This is easily seen by noting the maximal joint information between two such parameters is given by a Gaussian probability distribution in both, not a box distribution of width $2^{-n} L_0$ for some characteristic scale $L_0$!

Answer 5

The uncertainties in the relation point to inherent uncertainty. It's not because of our interaction that Nature is uncertain. This might be thought of when one reads some explanations of the relation. These explanations bring up the intervention of the observer with what is observed. It looks as if the observer brings about this uncertainty by disturbing some feature(s) he/she wants to know something about.
Of course, disturbance takes place. But the uncertainty relation posits that already before the actual observations are made there is uncertainty. You'll know beforehand that the quantities you are about to measure will be subject to this relation (if these quantities are "conjugate" quantities, i.e. the ones that appear in the UR).
There are experimental tests that verify the relation. These tests don't explain the relation, though (by saying that the uncertainties occur because of the physical intervention). For example this one.

So, how shall a teacher explain the HUR to her/his children?
Maybe like this:
If you make measurements of two quantities of a physical same system, then you'll observe, that for certain couples of quantities, these quantities can't have a precise value at the same time. These measurements have to be made at the same time, or shortly after another because otherwise the system will have evolved to a new state and in that case, the measurements can be said to two independent ones. Decoherence$^{*}$ will have taken place if you wait too long to make the second measurement. If the outcomes of the two measurements depend on the order in which you make them, then the quantities obey the uncertainty relation.

$^{*}$ Technically, decoherence is the failure of a state to stay in a superposition of states, which is, for example, an issue in the development of quantum computers. There a superposition of as many states is needed. But the more states are involved, the shorter the superposition will exist.