Heisenberg uncertainty - uncertainty about its meaning

[malawi_glenn]

So if I understand: there is x spin and x angular momentum; and y spin and y angular momentum. x spin and x angular momentum form a conjugate pair, correct? Am I saying this correctly?

i give up:P

 

[DrChinese]

i give up:P

 

[malawi_glenn]

you know J is not always L + S, J is also general angular momentum in SO(3). See for instance Sakurai.

We can specify to orbital angular momentum, L, L_x and L_y does not commute and we have an uncertainty relation for those two.

I would say that this is a quantum mechanical property, observables are assigned o quantized hermitian operators on a Hilbert space. If operators for two observables don't commute, there is an intrinsic limit for how good we can measure these two observables.

mn4j would say that this is just math, it has nothing to do with quantum mechanics, the uncertainty relation for L_x and L_y boils down to so(3) Lie algebra relation, nothing strange about it he would say (this is how how explained that HUP was not strange for position and momenta).

But quantum mechanics IS a MATHEMATICAL framework, to describe nature, so we should not be surprised that the properties of QM lies in it's math. Tautology I would say. The interesting thing is how this mathematical framework differs from the framework we choose for describing classical (Newtonian, non-quantum) mechanics.

 

[DrChinese]

... I would say that this is a quantum mechanical property, observables are assigned o quantized hermitian operators on a Hilbert space. If operators for two observables don't commute, there is an intrinsic limit for how good we can measure these two observables.

mn4j would say that this is just math, it has nothing to do with quantum mechanics, the uncertainty relation for L_x and L_y boils down to so(3) Lie algebra relation, nothing strange about it he would say (this is how how explained that HUP was not strange for position and momenta).

But quantum mechanics IS a MATHEMATICAL framework, to describe nature, so we should not be surprised that the properties of QM lies in it's math. Tautology I would say. The interesting thing is how this mathematical framework differs from the framework we choose for describing classical (Newtonian, non-quantum) mechanics.

Thanks. I can follow the idea that the standard formalism describes the world according to QM. (And I see your point about the tautology.) I don't follow the idea that those same ideas describe the classical world (and evidently you don't either, since you remark that the framework is different).

Thanks for your time.

 

[ajw1]

I must confess I do not understand everything that has been said in this thread, but I'm very interested in whether or not there is a demonstrable difference between classical and Heisenberg's quantum uncertainty (and of course, what this difference really is).

If I understand correctly the original reasoning from Heisenberg was quite 'classical', being in laymen language that fotons would disturb the particle to much to get a proper reading of it's position and momentum.

 

[DrChinese]

If I understand correctly the original reasoning from Heisenberg was quite 'classical', being in laymen language that fotons would disturb the particle to much to get a proper reading of it's position and momentum.

We are never satisfied unless we get thoroughly off-topic...

To answer: Heisenberg and others did use your analogy early on. They thought it made sense (which it sort of does) and made it easier to explain. However, with the advent of entanglement it has become clear that the HUP has nothing to do with the ordinary impact of a measuring device. If that was the issue, then entangled particles would not respect the HUP - and yet they do. You cannot use entangled particles to "defeat" the HUP.

I would recommend throwing any classical explanation out the window, and so would most physicists.

 

[ajw1]

I would recommend throwing any classical explanation out the window, and so would most physicists.

I would like to do that, but not without knowing why . Many basic physics books explain Heisenberg the way mn4j has interpreted it, being a natural result of an observable and it's derivate, and don't explain how this uncertainty becomes 'quantum'.

Can I conclude that the HUP is derived in a classical way, but that later experiments led physicist to give a different interpretation to HUP?

 

[malawi_glenn]

ajw1, if I measure my pencil position at my desktop (where it is at rest) then I can determine very very precisely it's position and momentum (it is at rest), so where is the uncertainty?

The thing one did is that since you have this uncertainty relation for Fourier pairs, i.e when you deal with waves, you also had DeBroigle waves for sub-atomic particles. That was how the early fathers of QM thought, that since we have wave mechanics, the position and momentum of the particles are wave-like, we should be able to ascribe the same uncertainty relation to them as for 'ordinary' waves.

The difference between classical physics and quantum physics is it's formalism. QM has wavefunctions, and operators and operation with those operators on the wavefunctions will give you your observables. This is not the way classical mechanics works.

Classical mechanics is deterministic, wheras quantum is probabilistic. Given the particle position and momentum at t = 0, you can at 100% accuracy determine where it will be located at t= T (in classical mechanics, note that this is in PRINCIPAL, in REALITY it is a bit more tricky, but that is not the point here - we want to discuss the internal properties, not the experimental difficulties and differences)

In quantum, you can not determine with 100% where the particle will be at t=T, since the dynamics is goverened by the Schrödinger equation, which takes your wavefunction (probability amplitude) from t=0 to t=T, i.e you measure that the particle is at x=0 at t=0, then the wave function has collapsed to a dirac-delta function, but as you let the particle go again, the wave function becomes spread out and you have NO IDEA where the particle will be at t = T. The only thing you can do, if you know what Hamiltonian is generating it's time evolution, is to calculate things like: it is 10% probability that I will find the particle between x= -X and x= X at t = T. And so on, you can never, in principle, say where it's going to be, even if we had PERFECT instruments, we can never say where it's going to be for sure.

 

[ajw1]

Thank you, malawi_glenn for your answer.

So what I thought is that the HUP itself somehow should 'prove' the quantum formalism. But apparently, as you seem to confirm, it is derived from (classical) wave mechanics.

 

[malawi_glenn]

Thank you, malawi_glenn for your answer.

So what I thought is that the HUP itself somehow should 'prove' the quantum formalism. But apparently, as you seem to confirm, it is derived from (classical) wave mechanics.

wave mechanics is not 'classical', waves are a physical framework. The wave nature of electron beams is NOT classical. One borrowed the formalism of waves in order to describe particles and systems on the microscopic level. Waves is a tool, a formalism, which can have applications. The bold idea of DeBroigle to describe subatomic particles with waves was brilliant and bold. Similarly did Einstein proposed to describe photons as particles/quanta on not waves.

Many things in QM are derived from classical mechanics, such as you replace the classical commutator (the Poisson bracket) with the quantum commutator which is the poisson bracket times i/hbar. And the hamiltonian is also the time evolution generator in quantum, just as in classical.

But the difference is that the physics is assigned to the indeterministic wavefunction, the hamiltonian evolve the wavefunction, wheras in classical physics this time evolution is deterministic.

So there are many analogies with quantum and classical mechanics, the great difference is that quantum is indeterminstic and classical is deterministic.

 

[jtbell]

If I understand correctly the original reasoning from Heisenberg was quite 'classical', being in laymen language that fotons would disturb the particle to much to get a proper reading of it's position and momentum.

The best way to view the HUP is as a statistical statement about the results of measurements of the quantities involved.

Imagine that you have many particles that are prepared identically so that they are in the same quantum state. You can imagine preparing them one at a time, if you like.

Take some of those particles and measure their position at a certain time after preparation. In general, you get a different result for each measurement, no matter what your measurement procedure is, or how precise your measuring instrument is. The standard deviation of those measurements is $\Delta x$.

Take some of those particles and measure their momentum at a certain time after preparation, the same time you used in the preceding paragraph. In general, you get a different result for each measurement, no matter what your measurement procedure is, or how precise your measuring instrument is. The standard deviation of those measurements is $\Delta p$.

The HUP says that for any state you put the particles in (no matter how you prepare them), and no matter how you measure their position and momentum, you will always find that

$$\Delta x \Delta p \ge \frac{\hbar}{2}$$

 

[jnorman]

the rhetoric contained within this thread notwithstanding, i have to state that much of this discourse goes against what i was taught about HUP. this is NOT just a mathematics problem, and it is not just the fact that we are unable to accurately measure both location and momentum simultaneously (since measuring one affects the other). from what i was taught, HUP indicates that, at the quantum level, particles truly do not have specific values for both momentum and location.

the AIP site on HUP indicates:
" Heisenberg concluded that the mutual uncertainties in position and momentum or energy and time really do exist. They are not the fault of the experimenter or the apparatus. They are a fundamental consequence of the quantum equations, built into every experiment in which quantum mechanics comes into play."

thus, it seems misleading to state that HUP is merely an indication of our inability to physically measure both properties simultaneously, or that such an interpretation is acceptable within QM. further, i do not think that just because some people remain confused on this topic, or for whatever reason do not agree on its interpretation, that the fundamental and primary interpretation by Heisenberg himself does not stand.

 

[ajw1]

So there are many analogies with quantum and classical mechanics, the great difference is that quantum is indeterminstic and classical is deterministic.

I do know about the non-deteministic interpretation of QM.

Allow me to refrase my question: does or doesn't the HUP itself contain proof for this non-deterministic interpretation, and , if so, in what way?

Or should we say it's derivation is just based on the wave behaviour of particles (without specifying the formalism of this wave), and other facts are needed to interprete this wave behaviour as being non-deterministic?

 

[malawi_glenn]

proof and proof, HUP is something you derive...

The indeterministic behaviour of QM is from experiments which leads to the postulates of QM (the copenhagen interpretation). HUP is something which you derive from the postulates of QM (just as length contraction is something which you derive from the postulates of SR)

HUP is, as jtbell very nicely explained, what you will obtain from a statistical survey, it is quite meaningless to speak about the HUP for one single particle. We are discussing probabilities here, never forget that.

 

[Ilja]

I do know about the non-deteministic interpretation of QM.

Allow me to refrase my question: does or doesn't the HUP itself contain proof for this non-deterministic interpretation, and , if so, in what way?

It doesn't. For the simple reason that there exists also a deterministic interpretation of QM, namely the pilot wave interpretation. In this interpretation, the HUP can be derived too.

 

[ajw1]

I was just also re-evaluating this statement from malawi, and I think it is not valid because for the pencil you pre-define its position and momentum. In this sense you have (in quantum terms) done a measurement, and the uncertainty has collapsed to a well known state.

The same is said to happen for particles when repeatingly measured.

The difference might be that the particle regains its uncertainty after a longer period between two measurements (without external influence), while we may assume that the pencil maintains its defined state. But this time-dependent evolution of the uncertainty after a measurement is not expressed in the HUP.

The argument from Ilja seems valid for me.

 

[malawi_glenn]

we don't have hbar in classical physics, you are mixing quantum and classical mechanics. Where does the hbar come into play when you do the 'classical uncertainty' realation??

You can only invoke hbar if you describe the pencil with QM

That is the error you both do, you look at the pencil as if it was a quantum system, I only consider the FORMALISM of classical mechanics vs. Quantum mechanics.

Of course your are correct, that in REALITY the pencil SHOULD be described by a wavefunction etc, but that was not my point, my point was that there is a difference in the FORMALISMS.

 

[yossell]

the rhetoric contained within this thread notwithstanding, i have to state that much of this discourse goes against what i was taught about HUP. this is NOT just a mathematics problem, and it is not just the fact that we are unable to accurately measure both location and momentum simultaneously (since measuring one affects the other). from what i was taught, HUP indicates that, at the quantum level, particles truly do not have specific values for both momentum and location.

In my experience, people say very different things about HUP - even within the same textbook, they shift their interpretation. Indeed, it's not quite clear what interpretation Heisenberg had - I've heard that he was pressurised into interpreting his principle as saying that particles lack specific values by Bohr, but that in his wider writings he seems to have a more epistemic interpretation - that the HUP represent an ineliminable lack of knowledge in some way.

" Heisenberg concluded that the mutual uncertainties in position and momentum or energy and time really do exist. They are not the fault of the experimenter or the apparatus. They are a fundamental consequence of the quantum equations, built into every experiment in which quantum mechanics comes into play."

thus, it seems misleading to state that HUP is merely an indication of our inability to physically measure both properties simultaneously, or that such an interpretation is acceptable within QM.

Some interpretations of HUP concern how the system will evolve: if you set up a system where an object is highly localised - so the spread of its possible values is small - and then perform a momentum measurement, then there will be a wide range of possible momenta values the object can have; such indetermism may be an objective feature of reality and, as such, it goes beyond indicating a mere inability to physically measure both properties simultaneously. So I think one could accept what you say here without having to commit oneself immediately to the stronger view that quantum objects actually lack values in certain situations.

Of course, there might still be strong reasons for thinking this - but those who wish to limit their discussion of HUP to expectation values can still think it represents an objective fact about the world without being drawn into delicate questions about fuzzy values...which is what they want.

 

[jtbell]

Tip: the closing tag for the blockquote must have a forward-slash: [/quote], not a backward-slash: [\quote]. You're probably thinking of LaTeX.

 

[duxz]

ehmm..hi.I thought the electrons themselves exist in innumerable probabilities of position and momentum.its a cloud of "possible" positions with every possibility having a possible velocity.i am just a school student just interested in physics.tell me if i am wrong-"you can't measure them, you can,t locate them but you know the possibilities of its p & x.so it might as well be considered aqs existing as a probability cloud!

 

[jtbell]

There is no generally accepted answer to your question. Some physicists believe that it doesn't make any sense to think of the particle's position and momentum as "existing" (in some sense) before they are measured or observed. Others think that it does.

Although it seems perfectly reasonable that, viewed all by itself, the position and momentum of a particle should always "exist," it turns out that versions of quantum mechanics that explicitly have this feature, have other features that many physicists find difficult to accept. As of now at least, no one has figured out a way (even in principle) to decide between these two positions by experiment. And so people argue about it a lot, for example in the threads here that have "Bohm" or "Bohmian" in the title.