Is uncertainity a postulate? [duplicate]

Question

I heard the standard interpretation of Heisenberg Uncertainty Principle: Just the measurement affects the position of the body because always you want to see a body (=to measure the position), you need a light - and only the fact that photons that strike the surface of that radiating body have momentum mean that the position of the radiating body isn't certain (even theoretically) only because of your measuring - and therefore the uncertainty in position will always be non-zero. Is there some possible way to modify this model to apply always in quantum mechanics? If no, is there some good model what describes the origin of the uncertainty?

Answer 1

Your example is probably not a good one to understand Heisenberg uncertainty with, because it mixes two uncertainty phenomena together:

1. The observer effect (See Wikipedia page of same name);
2. Heisenberg uncertainty itself.

The observer effect is the obvious and everyday observation that we can't extract information from a system without disturbing it in some way. We must change its physics ever so slightly, and the measurement of one variable will change measurement of another by some positive amount $\epsilon>0$. This differs from Heisenberg uncertainty in two ways: (1) We can in theory make the disturbance $\epsilon$ as small as we like by making better and better instruments with higher and higher signal to noise ratio; even though $\epsilon$ is always nonzero, there is no fundamental limit to how small it can be given a sophisticated enough experimental technique; (2) If we know the physics of our system perfectly and characterize our instruments properly, we can in principle calculate exactly what the observer effect is and thus calculate exactly what the measured quantities would have been were the instrument in question perfect.

The measurement of a voltage with a voltmeter, and the requirement that it needs to draw current from the measured circuit to make its measurement is a good example to model to explore these ideas with.

Heisenberg uncertainty, on the other hand, puts fundamental limits on the size of the uncertainty - you can't get it smaller than a certain limit. Moreover, the notion of measurement outcome having meaning before the measurement is made (the kind of notion that allows you to allow for the measurement effect above and state what the "true" measurement would have been) has been shown to be more and more untenable in modern research, in particular by experimental observation of the violation of the Bell inequalities See the "Counterfactual Definiteness" Wiki page. The jargon one hears in this context is that "violation of the Bell inequalities falsifies the notion of counterfactual reality".

To explain the fundamental limit, it is best to answer the question in your title, that is, how "fundamental" the HUP is. Physics is not "derived" from axioms as is mathematics, but there is a very real sense wherein the HUP is not axiomatic, namely, you don't need to include it in a reasonable list of basic postulates because it follows immediately from them. What are these postulates? Basically that quantum states live in Hilbert spaces and measurements are modelled by observables, which are self-adjoint operators acting on the state space together with a recipe for interpreting how these operators define the probability distributions of their measurments when a given system quantum state prevails. I explain this in more detail here (look in particular at the latter part of the answer where I describe observables). But now, given this QM in a nutshell, it immediately follows that whenever self adjoint operators do not commute (and the "usual" situation for matrices is that they do not), there is an uncertainly relationship between measurements modelled by the two operators. The most important noncommutation is that $[\hat{x},\,\hat{p}]=i\,\hbar\,\mathrm{id}$ between the position and momentum observables, leading to $\Delta x\,\Delta p\geq \hbar/2$ (where the $\Delta$s are standard deviations).

Now we live in a highly noncommutative world, so that on the one hand the uncertainty principle results directly from something as everyday and familiar as the notion that the shoe and sock on-putting operators do not commute when you get dressed in the morning. But on the other hand, whilst you can unscramble a warddrobe malfunction by undoing the respective operators and imparting them in the right order, observables do not map the quantum state - they only tell us how to calculate the measurement probability distribution, so they cannot in any sense be "inverted" and moreover, since the measurement by $\hat{x}$ co-indides with the system choosing a random eigenstate of the observable, information about what the measurement $\hat{p}$ would have been were it made instead is destroyed. Indeed, the experimental violation of the Bell inequalities suggests that the notion of "would have been" is meaningless.

Lastly, I do know that examples exactly like yours are given in the early chapters of the third volume of the Feynman Lectures to explain uncertainty. I admire the Feynman lectures greatly and would recommend the first eight chapters of volume 3 as an excellent introduction to the gist of quantum mechanics. But this particular example I don't think is the best, for the reason it mixes the two uncertainty phenomena as I stated at the beginning of my answer.

Answer 2

All physics theories depend on rigorous mathematics with its axioms and theorems, but in order to make a connection to measurements relationships between the mathematics and the physical observables have to be established, of equal importance to mathematical axioms, called laws or postulates, for the case of quantum mechanics.

I have found this list of postulates for quantum mechanics useful and common to most published lists:

The Postulates of Quantum Mechanics

1. Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system.

2. With every physical observable q there is associated an operator Q, which when operating upon the wavefunction associated with a definite value of that observable will yield that value times the wavefunction.

3. Any operator Q associated with a physically measurable property q will be Hermitian.

4. The set of eigenfunctions of operator Q will form a complete set of linearly independent functions.

5. For a system described by a given wavefunction, the expectation value of any property q can be found by performing the expectation value integral with respect to that wavefunction

6. The time evolution of the wavefunction is given by the time dependent Schrodinger equation.

As you see the Heisenberg uncertainty is not within the list. Nevertheless, as with theorems which arising from axioms can replace axioms, except that they are more complicated and simplicity is valued, the HUP arises from these postulates similar to a theorem , derived from the postulates using the mathematical tools.

It arises from postulate 1 and 2 and the commutator relationships that come out of the mathematical tools needed.

Actually there exists a reference on the web where the HUP is treated as a postulate.

Edit after comments: One has to clarify the use of the word "postulate" in physics and particularly in quantum mechanics theory, since a discussion at the comments claimed that a postulate is a synonym for an axiom. Even though the dictionary definitions are different, a google search shows that for mathematicians, a postulate and an axiom may be synonyms. In the treatment of quantum mechanics most rigorous expositions of an "axiomatic" derivation use "postulate" for the assumed "truths" that connect observable/measurable quantities to mathematical forms. These pick up a subset of the possible mathematical formulations , the subset that fits and predicts measurements.

Answer 3

The Heisenberg uncertainty principle is not an axiom in quantum mechanics. It has a legitimate derivation, based on the Schwarz Inequality in Linear Algebra. As 0celo7 pointed out in a comment, the axiomatic foundation of quantum mechanics (a la Dirac) takes the non-commutation of the position and momentum operators (${\hat x}$ and ${\hat p}$) as an axiom in the theory, viz. $\left[ {\hat x}, {\hat p} \right] = i \hbar$. In accordance with the inequality, the product of variances of non-commuting operators has a lower bound, dependent on the magnitude of the commutator. Thus, there would be an uncertainty principle for every pair of non-commuting operators (e.g. the angular momentum - angle, or the difficult Energy - time uncertainty relation).

As far as imagining how this implements out in practice is concerned, the example of momentum being imparted in the act of measurement is concerned is a convenient example, since it does put across the message. Please note however, that these interpretations sit atop the solid, mathematical foundation of the theory, which is reliant on a proper formalism. The mathematical formalism is based entirely on the language in which QM is written, which is linear algebra, since the theory treats the action of operators on basis vectors. There is absolutely no deficiency in this formalism which needs to be corrected, and the uncertainty principle is more of natural consequence of the non-commutation than a deficiency.

Answer 4

the measurement affects the position of the body

To be clear a so called measurement doesn't reveal some preexisting value, it is an interaction that happens continuously over time and leaves the resulting state in one of many fixed final states. Each of which has a real number associated with it.

mean that the position of the radiating body isn't certain (even theoretically) only because of your measuring

This isn't what the uncertainty principle is about. I have a good answer about what the uncertainty principle is at https://physics.stackexchange.com/a/169757 and an essential aspect is that the uncertainty principle relates two uncertainties. It says that if the order you do two so called measurements matters (if doing A then B then A again can give different results compared by on doing A twice then B which always has the second A give the same result as the first A) then there might be a lower bound on the products of the uncertainty of the two measurements. If the product of the two uncertainties has a lower bound and one of the uncertainties is low then the other one has to be high.

And that is the real affect of the uncertainty principle. It tells you about a trade off. You could choose to have a very low uncertainty for one kind of measurement but eventually when the uncertainty for that measurement gets low enough then if you keep decreasing the uncertainty of that one measurement then the other measurements that aren't compatible with that one can start to have high uncertainties.

And to be clear when o say decrease the uncertainty I mean get up and walk away from that lab and find a lab where the states are different states that have less uncertainty in that measurement.

This might still be confusing so let's talk about what uncertainty is. First you need a state. Lots of copies of the state, many systems prepared to be in the same state. Then you can measure a bunch of them. You get lots of results. They might have a mean and a standard deviation. Even better the sameple mean and sample standard deviation might come from a probability distribution with a population mean and a population standard deviation. That population standard deviation is what we call the uncertainty. It depended on the state as well as the thing you choose to measure.

The uncertainty principle isn't really talking about the uncertainty of that one measurement it is talking about multiple measurement of two different things, look at $$\Delta x\Delta p\geq \hbar.2$$

See how it has a $\Delta x,$ that's the population standard deviation of measuring the position $x.$ See how it has a $\Delta p,$ that's the population standard deviation of measuring the momentum $p.$ it says the product can't be too small.

The $\Delta x$ and $\Delta p$ can be small. You can pick a state where $\Delta x$ is say $10^{-20}m$ and you can pick a state where $\Delta p$ is say $10^{-20}kg m/s$ but you can't pick a state where both standard deviations are that small.

It says that way back when you picked a state there was a trade off. The results of one measurement could be made to have a small standard deviation and the results of the other standard deviation for the same state would then have to be larger.

Sometimes both standard deviations are large. That happens for some states. But they can't both be small. Another thing that can happen is that a state can fail to even have a standard deviation or even a mean. This happens for instance if one the measurements has a zero standard deviation then the other one won't even have a standard deviation. There is nothing wrong with not having a mean and not having a standard deviation. However it is worth checking whether what you proposed can be done done in the lab without using an infinitely big piece of equipment or one that can create arbitrarily high energies.

the uncertainty in position will always be non-zero.

The uncertainty in position can be zero. But this requires arbitrarily high energies and the momentum distribution has every momentum equally likely from $-\infty$ to $+\infty$ and so the momentum uncertainty is in some sense infinity because every momentum is equally likely. It has no mean and so you can't even define the standard deviation (since that depends on the mean). In a loose sense the standard deviation is infinite.

Similarly the uncertainty in momentum can be zero. But this requires am infinite space for your particle to spread out in. The position distribution has every position equally likely from $-\infty$ to $+\infty$ and so the position uncertainty is in some sense infinity because every position is equally likely. It has no mean and so you can't even define the standard deviation (since that depends on the mean). In a loose sense the standard deviation is infinite.

The real point is that when to choose states with a real low uncertainty in one thing that same state ends up having a high uncertainty on another thing. That's the uncertainty principle.

Is there some possible way to modify this model to apply always in quantum mechanics?

You can generalize the uncertainty principle to any two measurements. If the two measurements are compatible (i.e. A then B then A then B again always gives the same results the a b a b in the sense that you get the same thing for your A measurement the second time and get the same thing the second time for your b measurement) then the uncertainty in each can be as low as you want.

Otherwise there is a lower bound on the two. However the lower bound actually depends on the mean of something else. So even when they aren't compatible sometimes they can still both be small. That doesn't happen for position and momentum since the thing whose mean you want is something that always has the same mean for every state.

If no, is there some good model what describes the origin of the uncertainity?

Yes. The Schrödinger equation has enough information to show you exactly where uncertainty comes from. And you can derive the uncertainty principle from it.

So see uncertainty a cheap way is to just look at a wavefunction and its Fourier transform and note that each gives the distribution for the position and momentum measurements respectively. So then you get the quantum uncertainty from the acoustic one, that to have a chirp you need lots of pitches and to have a good pitch you have to have a long tone.

But that jumps over how you get probability distributions from a state. The full answer for that requires answering the measurement problem (which you did tag so I'll mention it).

I'll talk about the simplest measurement, a Stern-Gerlach measurement of a spin 1/ 2 particle. It is the simplest because it only gives two possible result and so it always has a mean and a standard deviation.

And the results can be computed with the Schrödinger equation and nothing else. Just write down the Hamiltonian for an inhomogeneous magnetic field inside the device.

So you have a device where beams come in and there are two places beams can come out. An up one and a down one. If you have three devices you could bolt one down and then put one of the others at each the two output ends so the output of the bolted one is the bin out of the other one.

If you do this then all the ones that go in the up box come out of the up of that box and all the ones that come out the down box come out the down of that box.

This is an example of multiple copies of the device being compatible with itself measuring with it twice always the same result the second time as you got the first time. So up the first time means you get up the second time. And down the first time means you get down the second time.

The device usually splits an incoming beam into two beams one going up and one going down. But it changes the result for each outgoing beam so that it becomes one of the special beams that doesn't get split and comes out just one end.

Why do we say it changes things? Because if we put three in a row with the second on the up output of the first and the third in the down output of second then the third one never has anything come out (once it went up it always went up). So the output of the first box really is a thing with the property that it always goes up.

But if you rotate that second box so it still gets the up output of the first box but now sends the beams left and right then no matter which one (left or right) you attach the third box to then beams come out of both the up and the down parts of the third unrotated box. So that property of always giving up was destroyed by the second box. So the rotated devices are incompatible with each other. And since the output of the first box had the property that it gives the same result (up) and then didn't have that property after it went through the second box, so the second box destroyed that property so changed the particle.

Every device changes a particle unless it happens to already be in one of the few kinds of output states that device allows.

So we know that so called measurement changes things. And it changes things into one of the few states that that device doesn't change. And if there isn't a state that both devices leave alone then the order you do them matters since they will change it back and forth. Like of you had two remotes for a TV and two people that really want to make the TV be on different stations. Compatible results would be like someone that only cares about volume and the other cares about the station eventually you get to a state that neither will change.

So we know that we can get more than one result. What about the standard deviations. For that you need to know the relative frequency of getting each result. We have two results. Now you can put s detector in front of the incoming beam and it will fire at a certain rate. That rate depends on the current density and the width of the beam, so depends on the current. Each separate beam has its own current density and its own width. And the Schrödinger equation accurately predicts the current density at each point (it predicts the whole wave from which you can get the current density at each point) so you can get the current.

So just writing the initial input state and the Hamiltonian for the device and then evolving with the Schrödinger equation gets you the rate rate a detector at any device location fires. There are shortcuts. The relative size of the square of the projection of the spin state onto the eigenstates of an operator tells you the relative frequencies of the rates.

And you can predict how the detector works with the Schrödinger equation. Bit it works the same way, by splitting the wave but now you have two things the particle and the detector so we have to be honest that for the Schrödinger equation the wave is in configuration space. So the split happens in configuration space. And what happens is that originally just the position and spin on the particle changed then the detector changes too and you end up having two groups of configurations that each acts like they are the only group. And forever will. These effectively separate worlds can have other devices that count the relative frequencies of repeated measurements and the group of configurations will become almost entirely focused on a reading near the population average. And since detectors have to have a certain insensitivity to noise from other sources they are also insensitive to the much smaller group of configurations where it isn't nearer the average.

This is similar to a statistical effect of the law of large numbers. It isn't the law of large numbers since there isn't a fixed sample space (remember how measurements change things, this means no foxed sample space so regular probability theory does not apply).