Physical justification for Uncertainty principle

[loom91]

Most popular or elementary textbook level expositions advance the following physical argument in favor of the uncertainty principle:

In order to observe a state we must disturb it. Thus we have changed the state by our very act of observation and uncertainty creeps in.

Now this explanation is obviously wrong to me for more than one reasons, but what is the real explanation. Does an explanation exist or is it taken as an unexplainable axiom? Thanks.

Molu

 

[dextercioby]

It's a logical consequence of the 5+1 axioms of QM.

Daniel.

 

[lalbatros]

loom91,
dextercioby,

Don't stick too much to the 'axioms' or to the 'observation' stuff.
The wavepacket picture is in itself already very clear and much simpler:

if a wavepacket has a sharp localisation it cannot have a sharp wavevector spectrum, and vice-versa​

This picture covers only a small part of the application of the uncertaincy principle.
But it show that the "axiom" or "observation" point of views are not absolutely necessary.

However, there is a link. Of course, if you want to 'prepare' or to 'choose' a system a wave-system that has some localisation in space, then you know that -because it is a wave system- you will have an undefinite wavevector.

Now, there is still something to clarify: the relation between momentum and wavevector. Saying that this is simply Planck's relation tells us rather little, except that a universal constant is involved. But where comes this relation from, physically. What is its interpretation? There is none I think, only that QM is more general than CM.

I find fascinating how Euler, Lagrange, Hamilton, and others were able to put classical mechanics in the so-called 'canonical form' without knowing anything about the quantum. This is so impressive that I feel guilty for not being able to explain the path from momentum to wavevector.

Michel

 

[DaveC426913]

Loom, I don't know what your level of knowledge on the subject is, but mine is way below the level of the two above answers. They're of no help to me. I'll attempt to answer it in layperson's terms. Forgive me if I underestimate your level of knowledge.

Most popular or elementary textbook level expositions advance the following physical argument in favor of the uncertainty principle:

In order to observe a state we must disturb it. Thus we have changed the state by our very act of observation and uncertainty creeps in.

Now this explanation is obviously wrong to me for more than one reasons, but what is the real explanation. Does an explanation exist or is it taken as an unexplainable axiom? Thanks.

Molu

First, be careful of the term "obvious". Nothing should be obvious. We're dealing with atomic and subatomic particles here, something that cannot be compared to macroscopic objects like rocks and people.

Second, be careful of the assumptions you make when talking about "obseving" something. In the macro world, we use an observing medium (light) that is WAY smaller than the objects we are trying to observe - it has no noticeable effect on them. But in the atomic world, our observing methods are on the scale of the things being observed.

Consider what would happen if you scaled your atomic observations up to macro size. Say you wanted to observe the speed an position of a grapefruit-sized atom across the table. Not too hard to do - unless your only source of information about its speed and position was by pelting it with apples! Further, you don't get to actually SEE the apples OR the grapefruit, all you get to do to record where the APPLES stopped after they bounced.

What's worse, slow-moving apples do a lousy job of telling you where the grapefruit is, or is going. Faster-moving apples will give you a better idea, but how do you think these faster-moving apples will affect the velocity and position of the grapefruit? It'll knock the grapefruit around even MORE.

Now, after the first few apples have ricocheted off the grapefruit, do you think it still has the same position or velocity?

 

[abszero]

The physical justification is that it works.

Dave, what you've said would imply that the Uncertainty Principle is a purely experimental phenomena. In fact, it is absolutely necessary for our current formulation of quantum mechanics, and is a consequence of the formal structure, not just some appendix to it.

What the OP said is more or less correct, to observe we must disturb, and unlike in classical mechanics, in quantum mechanics there is no omniscient observer measuring every dynamical quantity. The observer is part of the system, so the order of measurement that an observer chooses directly affects the system being measured, and there's not way around it.

 

[ZapperZ]

Second, be careful of the assumptions you make when talking about "obseving" something. In the macro world, we use an observing medium (light) that is WAY smaller than the objects we are trying to observe - it has no noticeable effect on them. But in the atomic world, our observing methods are on the scale of the things being observed.

Consider what would happen if you scaled your atomic observations up to macro size. Say you wanted to observe the speed an position of a grapefruit-sized atom across the table. Not too hard to do - unless your only source of information about its speed and position was by pelting it with apples! Further, you don't get to actually SEE the apples OR the grapefruit, all you get to do to record where the APPLES stopped after they bounced.

What's worse, slow-moving apples do a lousy job of telling you where the grapefruit is, or is going. Faster-moving apples will give you a better idea, but how do you think these faster-moving apples will affect the velocity and position of the grapefruit? It'll knock the grapefruit around even MORE.

Now, after the first few apples have ricocheted off the grapefruit, do you think it still has the same position or velocity?

I will have to concur with the previous post. You are implying that the HUP is a "measurement uncertainty". This is not correct. If it is, then does that mean that if I have a better instrument, I can actually change the HUP? The uncertainty that one gets in a single measurement, which is contained in what you have described above, is not the HUP.

Again, take note that the HUP is a natural consequence of the formulation and not a consequence of measurement instrumentation. The formulation provides no way around it.

Zz.

 

[DaveC426913]

Yes, it was not meant to be my implication that it was merely a measurement thing.

What I was trying to get across is that there is no way of making an observation without affecting the thing being observed.

 

[masudr]

Is the canonical comm. relation one of dextercioby's "5+1 postulates"?

 

[jbusc]

Here's a different formulation of Heisenberg based on Fourier analysis:

Consider the quantum wavefunction as describing the probability that you will find the particle at that location. Take the Fourier transform of the wavefunction.

Suppose you have determined the position of the particle to great accuracy. The quantum wavefunction then "looks" like a spike at the location where the particle is and zero elsewhere. What does the Fourier transform look like? It's all over the place since the range of frequencies that must be added up to get the wavefunction is huge - so the wave number of the quantum wavefunction is gigantic. Since wave number is related to momentum, position and momentum therefore are mutually uncertain.

Similarly, suppose you have determined wave number(i.e., momentum) to great certainty. The Fourier transform of the wavefunction is then a spike, indicating that the wavefunction has few frequency components - so the quantum wavefunction is all spread out, meaning there is uncertainty in position.

Finally, some knowledge of Fourier analysis says the minimum uncertainty happens when the quantum wavefunction is a gaussian. Therefore, the wave number is also a gaussian, and the product is 1/2

This can be converted into the familiar form by plugging in the momentum in place of the wave number, and saying that is the absolute minimum (which is where the greater-than or equals comes in)

 

[selfAdjoint]

Here's a different formulation of Heisenberg based on Fourier analysis:

Consider the quantum wavefunction as describing the probability that you will find the particle at that location. Take the Fourier transform of the wavefunction.

Suppose you have determined the position of the particle to great accuracy. The quantum wavefunction then "looks" like a spike at the location where the particle is and zero elsewhere. What does the Fourier transform look like? It's all over the place since the range of frequencies that must be added up to get the wavefunction is huge - so the wave number of the quantum wavefunction is gigantic. Since wave number is related to momentum, position and momentum therefore are mutually uncertain.

Similarly, suppose you have determined wave number(i.e., momentum) to great certainty. The Fourier transform of the wavefunction is then a spike, indicating that the wavefunction has few frequency components - so the quantum wavefunction is all spread out, meaning there is uncertainty in position.

Finally, some knowledge of Fourier analysis says the minimum uncertainty happens when the quantum wavefunction is a gaussian. Therefore, the wave number is also a gaussian, and the product is 1/2

This can be converted into the familiar form by plugging in the momentum in place of the wave number, and saying that is the absolute minimum (which is where the greater-than or equals comes in)

If I am not mistaken this was Heisenberg's original justification for his principle.

 

[jbusc]

If I am not mistaken this was Heisenberg's original justification for his principle.

I'm not exactly sure if it was his original, but I remember reading that Heisenberg initially fell into the same wrong conclusion, that it is error introduced by observation, and was convinced by Born (or possibly Bohr?) to look at his work again, after which he determined it was really a fundamental uncertainty.

 

[koantum]

In order to observe a state we must disturb it. Thus we have changed the state by our very act of observation and uncertainty creeps in. ...Does an explanation exist or is it taken as an unexplainable axiom? Molu

Molu, uncertainty has nothing to do with disturbances. To see this, we just have to remember that the quantum-mechanical probability algorithm is time-symmetric. It allows us to assign probabilities not only to the possible outcomes of future measurements (on the basis of actual outcomes of present measurements) but also to the possible outcomes of past measurements (on the same basis). (In both cases we use the Born rule. We can even assign probabilities to the possible outcomes of present measurements on the basis of past and future measurements if we use the Aharonov-Bergmann-Lebowitz rule instead.)

Suppose that a measurement at t₁ yields the value v₁ and a measurement at the later time t₂ yields the value v₂. If the measurements are of the repeatable kind and the Hamiltonian is 0, then according to the usual time-asymmetric interpretation of the formalism, the observable measured possesses the value v₁ from t₁ to t₂, at which time its value changes to v₂. If this is a valid interpretation than so is the following: the observable measured possesses the value v₂ from t₁, at which time its value suddenly changes from v₁, till the time t₂. If the latter interpretation is harebrained, then so is the first. If there is no collapse backward in time, then there is no collapse forward in time. And if there is no collapse, the question of disturbance does not arise. Observables have values only if, when, and to the extent that they are measured.

If you want to say something half testable about the values of observables between measurements, you can say it in terms of the probabilities of the possible outcomes of unperformed measurements.

 

[koantum]

It's a logical consequence of the 5+1 axioms of QM.

Daniel, right you are, but there is nothing sacrosanct about those axioms. You can choose others as long as they give you the same probability assignments. These days, post string theory and Susskind's landscape, anthropic arguments are again en vogue. Quoting Jeffrey Bub,

the fact that we find ourselves in a quantum world where measurement is possible... will surely involve the same sort of explanation as the fact that we find ourselves in a world where we are able to exist as carbon-based life forms.​

In this spirit, require the existence of objects that

• have spatial extent (they "occupy" space),
• are composed of a (large but) finite number of objects that lack spatial extent,
• are stable—they neither collapse nor explode the moment they are formed,

and find that a precondition for the existence of such objects is the fuzziness of both the relative positions and the relative momenta of their components, as I have explained in https://www.physicsforums.com/showthread.php?t=116582". For a stable equilibrium between

• the tendency of interatomic relative positions to become less fuzzy due to electrostatic attraction and
• the tendency of interatomic relative positions to become more fuzzy due to the fuzziness of the corresponding momenta,

we need a relation between Delta x and Delta p such that a decrease in one implies an increase in the other. It stands to reason that Delta p goes to infinity as Delta x goes to zero. A sharp position then implies a maximally fuzzy momentum (that is, a flat distribution containing no information whatever). This suffices to derive the free particle propagator and the free Schrödinger equation. Once we have that, the possibilities of incorporating interactions into the resulting formalism are very limited and largely determined by requiring the existence of measurement devices or, in other words, the consistency of the formalism, inasmuch this presupposes the existence measurement devices.

 

[koantum]

something to clarify: the relation between momentum and wavevector. Saying that this is simply Planck's relation tells us rather little, except that a universal constant is involved. But where comes this relation from, physically. What is its interpretation? There is none I think, only that QM is more general than CM.

Michel, the concepts that feature in classical physics have their roots in the mathematical features of the quantum-mechanical probability algorithms. The classical concepts of energy and momentum are rooted in the time and space dependence of the phases of quantum-mechanical probability amplitudes. (These amplitudes are complex numbers. While a real number has an absolute value and a sign, a complex number has an absolute value and a phase, which is a real number.)

 

[loom91]

Dave,

The explanation you give, a highly popular one first given by Heisenberg himself, is nevertheless a wrong one as pointed out by others and as I said myself. In fact post-EPR it is my understanding that it is entirely possible to avoid the observer effect. If Bob measures one of two entangled particles, he is also in fact measuring Alice's particle without in any way 'disturbing' Alice's particle. Because of this, any uncertainty principle constructed on the basis of observer effect would hold at classical mechanics but break down when the Bell Inequality is violated.

Molu

 

[loom91]

Here's a different formulation of Heisenberg based on Fourier analysis:

Consider the quantum wavefunction as describing the probability that you will find the particle at that location. Take the Fourier transform of the wavefunction.

Suppose you have determined the position of the particle to great accuracy. The quantum wavefunction then "looks" like a spike at the location where the particle is and zero elsewhere. What does the Fourier transform look like? It's all over the place since the range of frequencies that must be added up to get the wavefunction is huge - so the wave number of the quantum wavefunction is gigantic. Since wave number is related to momentum, position and momentum therefore are mutually uncertain.

Similarly, suppose you have determined wave number(i.e., momentum) to great certainty. The Fourier transform of the wavefunction is then a spike, indicating that the wavefunction has few frequency components - so the quantum wavefunction is all spread out, meaning there is uncertainty in position.

Finally, some knowledge of Fourier analysis says the minimum uncertainty happens when the quantum wavefunction is a gaussian. Therefore, the wave number is also a gaussian, and the product is 1/2

This can be converted into the familiar form by plugging in the momentum in place of the wave number, and saying that is the absolute minimum (which is where the greater-than or equals comes in)

I think that's the best explanation I've seen here. It explained things at the right level for me. Thanks :) Though I'm afraid I didn't take the trouble to understand Fourier Analysis, I can get the essence of the argument.

 

[Hitokiriilh]

There's actually not much magical stuff to the whole "uncertainty principle." "It comes from the axioms of QM." Well...it can...but you can also get it just from arguments involving the Fourier transform of a finite wave pulse. As such it can just as easily be obtained classically as it can quantum mechanically.

Even classically, if you look at a pulse of light and take its Fourier transform, you'll find that it behaves as though it has a distribution of frequencies with an average frequency that corresponds exactly to the frequency that normally appears in Planck's E=h*(nu). If you then say that with each frequency there is an associated energy of the same form as above, then you can obtain, via classical arguments, the "uncertainty principle." It's not really an uncertainty principle, per se, since this is classical, but you can find that the length of the time pulse and said energy distribution obeys exactly the same inequality. See? Oooooo! QM from classical! Of course the wave for "particles" requires quantum mechanics, but the principle can be obtained in a similar way...

If you want a better feel for the uncertainty principle, screw elementary texts and ad hoc hand-waving explanations like the one you mentioned and study Fourier transforms and some wave theory. I always found that elementary texts just made physics more confusing by replacing valid mathematics with the suspicious invocation of intuition and ad hoc explanations anyway.

EDIT: Oh! It looks like someone got to the Fourier transform point before I did. I would still just like to emphasize that in that argument you already get the jist of, there was NO mention of quantum mechanical phenomena aside from the wave function associated with the "particle." I think this begs the question: what other supposedly quantum mechanical phenomena can be obtained from classical arguments? You'd be surprised how much. Think about it some. Whenever you encounter something funny in quantum mechanics, ask "can this be obtained with equivalently classical methods?" There's been a lot of work in stochastic mechanics by people like Nelson and Boyers that show that classical particles subject to stochastically varying potentials or fields appear to obey quantum mechanical laws. Thought provoking, no?

Sorry to get off topic.