Can Quantum Mechanics Explain the Uncertainty Principle?

[Archosaur]

Hi all,

My understanding of the Heisenberg uncertainty principle is this:
1) in order to measure something, we have to hit it with something smaller.
2) in order to know the exact position of something we have to know the exact position of the thing we hit it with.
3) therefore, we cannot know the exact position of anything, because we don't and can't know the exact position of any smaller particle/wave we could measure it with.

My first question is: is this an okay explanation?

My second question is more general, and is my biggest issue with QM (so far).

Many people suggest the following leap in logic, but I can't follow it.
They say, since the HUP states that we cannot know the exact position/velocity of a particle, that a particle doesn't have one exact position/velocity.

This leap between position/velocity being "unknowable" by us and being actually "indeterminate" is lost on me. Just because you don't know how many fingers I'm holding up doesn't mean I'm not holding up a particular number of fingers. Sure, for your purposes, you might create a probability curve to represent the number, but, again, that doesn't mean I actually have a finger "cloud".

Help me get on track here.
Thanks.

 

[DaveC426913]

"They say, since the HUP states that we cannot know the exact position/velocity of a particle, that a particle doesn't have one exact position/velocity."

That's not the reason its position is not knowable.

HUP does state that the position/velocity of a particle is not simply a measurement problem, it is a real property.

Here's how I rationalize it. Whether it's a valid analogy or not, I'll leave to others:

I envision a particle as the sum-total of a wave packet smeared over a distance.

How do you define the exact location of a wave smeared over a distance? Pick the middle point? The narrower you try to pin down an exact location, the more of the wave you have to ignore.

Conversely, if you try to determine its total momentum, you must include the entire wave. If you count the entire wave, smeared over a distance, it becomes meaningless to talk about its exact location.

 

[twofish-quant]

My first question is: is this an okay explanation?

No it's not.

A better explanation is this...

Imagine a waves hitting seashore. Those waves have a well defined frequency, but not a well defined position.

Now imagine one giant wave hitting the shore. That one wave has a well defined position, but not a well defined frequency.

It so happens that when you do QM, the momentum of a particle corresponds to its frequency. So if you localize the particle in one place, it no longer has a well defined frequency and hence no well defined position.

Replace that ocean wave with a light wave, and you see the problem.

This leap between position/velocity being "unknowable" by us and being actually "indeterminate" is lost on me.

That's because you are using a wrong mental model. It turns out that particles have properties that are indeterminate rather than merely unknowable. It gets worse, since you can do a thought experiment to show that if particles actually had unknowable information, that all the particles in the universe would have to exchange information to keep the bookkeeping right.

 

[Feldoh]

The Heisenberg Uncertainty Principle is a mathematical property of QM.

You've got it backwards: "They say, since the HUP states that we cannot know the exact position/velocity of a particle, that a particle doesn't have one exact position/velocity."

The particle doesn't have one exact position/velocity therefore we cannot know the exact position/velocity of the particle. This is how you should look at the uncertainty principle.

 

[jambaugh]

The critical distinction between classical and quantum mechanics is that in QM we move from a language of system state to a language of observer-system interaction. This is why we go from "knowledge of system state" to "indeterminacy of system observable". It's a matter of no longer speaking in terms "what is" (ontological language) and speaking rather in terms of "what happens" (empirical action language or epistemological language).

As Orwell points out we think with language. We find this action language of QM more general allowing us to conceive of systems where the assumption that we have acted to determine one observable necessitates the indeterminacy of another observable.

It is more a matter of making explicit the implicit assumption that we have made measurements when we speak of the system's state. In making them explicit we must then also ask if the assumption that we've measured all variables is valid which raises the question of compatibility between different observables (such as position and momentum).

Once we see in the theory that the order in which such pairs of measurements are made matters we can no longer say a prior measurement (of say momentum) reflects current information about what we would see if we repeat the measurement given we have since made an incompatible measurement (of say position).

You could say measuring the position changed the momentum but that is using a language of classical states. Sticking to the language of actions in QM we should only say that measuring the position changes our ability to predict future measurements of momentum.

Being thus careful about format you avoid asking questions which cannot be answered by empirical means e.g. "do electrons have souls?". Once you are clear on the answers to such empirical questions (and have learned to make the distinction as to type of questions) you can go back and ask if and in what way ontological statements about system states can be made which are consistent with the empirical questions and their answers. You can then ponder hidden variables, EPR experiments and Bell type inequality violation.

 

[Demystifier]

Archosaur, you are absolutely right. If we cannot know something, it does not imply that it does not exist. Indeed, there is a formulation of quantum mechanics in which particles have both positions and momenta even when we don't measure them. This formulation is known as - the Bohmian interpretation.

 

[mccoy1]

I like the analogy very much and I dislike the word "meaningless" by the same margin.

 

[conway]

Archosaur, you are absolutely right. If we cannot know something, it does not imply that it does not exist. Indeed, there is a formulation of quantum mechanics in which particles have both positions and momenta even when we don't measure them. This formulation is known as - the Bohmian interpretation.

I have to admit I'm asking this question rhetorically because I have very little faith or expectation that anyone could answer it in a way that would satisfy me but in any case here it is: why don't your electrons radiate? The charge cloud doesn't radiate because it is stable, but what about your little particles being wafted about by the wave potential??

 

[Demystifier]

I have to admit I'm asking this question rhetorically because I have very little faith or expectation that anyone could answer it in a way that would satisfy me but in any case here it is: why don't your electrons radiate? The charge cloud doesn't radiate because it is stable, but what about your little particles being wafted about by the wave potential??

I have to admit I'm answering this question rhetorically because I have very little faith or expectation that anyone could answer it in a way that would satisfy you but in any case here it is:
When classical equations of motion (for charges and electromagnetic fields) are satisfied, then accelerated charges necessarily radiate. However, the Bohmian laws of motion are not the classical ones. The quantum potential affects not only the electrons, but photons as well. It turns out that this influence is such that it completely prevents radiation when the electron wave function is an energy eigenstate. In other words, my little particles being wafted about by the wave potential are stable too.

 

[jambaugh]

Hi all,
This leap between position/velocity being "unknowable" by us and being actually "indeterminate" is lost on me. Just because you don't know how many fingers I'm holding up doesn't mean I'm not holding up a particular number of fingers. Sure, for your purposes, you might create a probability curve to represent the number, but, again, that doesn't mean I actually have a finger "cloud".

I think the error people make is in confusing the "cloud" which represents our knowledge and uncertainty about how many fingers we would see if we look, for a representation of the realty of your fingers.

Begin with the interpretation of the wave function as a representation of knowledge about the system in the same sense that a probability distribution is. Understand that in QM when we speak of "an electron with momentum p" this is not a direct statement about the electron's state but a statement that we have made a momentum measurement of the electron, or more precisely that we know what the value of an immediate momentum measurement will be.

It is in this sense that we speak of indeterminacy of values of observables. This again is not a statement about the state of the system (that it's fuzzy) nor do we assume the system state has achieved some fuzzy cloud state. One is explicitly avoiding any reference to the system's state and only to what we know about how the system will behave with regard to potential future measurements. That is what is cloudy. It is not a matter of unreality but of agnosticism about reality.

It is fine to further speculate about the nature of the reality behind the predictions and descriptions in quantum theory. But you need to acknowledge that in so doing you are beginning to walk outside the actual theory which concerns itself only with what is an can be known about a physical system.

You can compare it to an ideal court trial where one can only infer facts based on admissible evidence. (Not a perfect analogy but another example of where one limits scope for the sake of rigor.)

Demystifier advocates a Bohmian pilot wave interpretation which reifies (treats as real) the wave-function. Others advocate Everett's many universes which assert a different universe exists for each possible combination of measurements. I see such further interpretation as futile as it is speculation about that which nothing can be empirically verified.

It is similar to aether theories of light propagation in classical relativity. You can introduce an aether and get all the same predictions of SR but there is no need and one is introducing that about which nothing can be empirically verified. The standard theory excises the aether and speaks only about that which we can know something. Hence time is what a clock reads and distances are what measuring rods measure. In QM system variables are outcomes of system measurements. We need only concern ourselves with how such measurements behave.

 

[conway]

The quantum potential affects not only the electrons, but photons as well. It turns out that this influence is such that it completely prevents radiation when the electron wave function is an energy eigenstate. In other words, my little particles being wafted about by the wave potential are stable too.

Do your electrons orbit the proton in little circles like the Bohr atom or do they meander randomly about?

 

[twofish-quant]

Archosaur, you are absolutely right. If we cannot know something, it does not imply that it does not exist. Indeed, there is a formulation of quantum mechanics in which particles have both positions and momenta even when we don't measure them. This formulation is known as - the Bohmian interpretation.

True and that works. The trouble with the Bohm interpretation is that it's a non-local theory. To get it to work you have to assert particles are secretly exchanging information instantaneously. What happens is that you can set up situations in which you have two particles whose quantum states are correlated, so if you measure one particle, somehow the information on how you measured the first particle influences the measurement you get in the second particle. (I'm wildly oversimplifying, but look up the Bell inequality for a specific situation that this happens.)

It's one of those "choose your poison situations." Something weird is going on.

 

[twofish-quant]

Demystifier advocates a Bohmian pilot wave interpretation which reifies (treats as real) the wave-function. Others advocate Everett's many universes which assert a different universe exists for each possible combination of measurements. I see such further interpretation as futile as it is speculation about that which nothing can be empirically verified.

One reason this is worth thinking about is that it's not immediately obvious that there isn't some sort of empirical difference between the different interpretations of quantum mechanics. One problem with the standard Copenhagen interpretation is that the interpretation doesn't very clearly define when a wave function collapse occurs

 

[Demystifier]

Do your electrons orbit the proton in little circles like the Bohr atom or do they meander randomly about?

The trajectory depends on the wave function and on the initial particle position, but it is certainly not random.

 

[Demystifier]

True and that works. The trouble with the Bohm interpretation is that it's a non-local theory. To get it to work you have to assert particles are secretly exchanging information instantaneously. What happens is that you can set up situations in which you have two particles whose quantum states are correlated, so if you measure one particle, somehow the information on how you measured the first particle influences the measurement you get in the second particle. (I'm wildly oversimplifying, but look up the Bell inequality for a specific situation that this happens.)

It's one of those "choose your poison situations." Something weird is going on.

That is all true, except that I don't see this as a trouble and I don't find it weird. But that's just me.

 

[zonde]

Many people suggest the following leap in logic, but I can't follow it.
They say, since the HUP states that we cannot know the exact position/velocity of a particle, that a particle doesn't have one exact position/velocity.

I can't follow it either. If there are billiard balls they have exact position and velocity even if they are very small.
So to me it seems that this statement can have alternate conclusion like that:
Since the HUP states that we cannot know the exact position/velocity of a particle, that particle is not a particle at all but wave instead.

 

[Codexus]

Since the HUP states that we cannot know the exact position/velocity of a particle, that particle is not a particle at all but wave instead.

Except those "waves" also show particle-like behavior in some experiments. So they are something that has a particle aspect and a wave aspect. Now maybe we should have invented a new name to avoid confusion with classical particles, but when we refer to particles in a QM context we're really talking about these wave/particles. It's really just a question of semantics.

Anyway they are weird things that we have trouble imagining since the world we experience in our everyday lives has nothing similar.

 

[zonde]

Except those "waves" also show particle-like behavior in some experiments. So they are something that has a particle aspect and a wave aspect. Now maybe we should have invented a new name to avoid confusion with classical particles, but when we refer to particles in a QM context we're really talking about these wave/particles. It's really just a question of semantics.

Anyway they are weird things that we have trouble imagining since the world we experience in our everyday lives has nothing similar.

Yes the waves are quantized. So probably the name "quanta" is the one you are looking for.
And clearly it could help imagining those things if we would find satisfactory example of quantized waves in macro world.

 

[DanP]

I can't follow it either. If there are billiard balls they have exact position and velocity even if they are very small.
So to me it seems that this statement can have alternate conclusion like that:
Since the HUP states that we cannot know the exact position/velocity of a particle, that particle is not a particle at all but wave instead.

I so believe that part of this confusion arise from the fact many high schools teachers like to state things like "photons are waves and particles at the same time". You'll have to simply step out from this mentality. Ill quote R. Feynman

Historically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither. Now we have given up. We say: "it is like neither".

Its a new animal. Something unlike anything you ever seen.

 

[conway]

The trajectory depends on the wave function and on the initial particle position, but it is certainly not random.

If you can't give me an example of a typical trajectory for an electron in a hydrogen atom, then I don't know what your theory is good for.

 

[DanP]

If you can't give me an example of a typical trajectory for an electron in a hydrogen atom, then I don't know what your theory is good for.

1s orbital. There, you have your answer.

 

[Fra]

Here is a simple idea to help make this plausible.

Just because you don't know how many fingers I'm holding up doesn't mean I'm not holding up a particular number of fingers. Sure, for your purposes, you might create a probability curve to represent the number, but, again, that doesn't mean I actually have a finger "cloud".

Right, but it means that the information I (as an observer) have about your hand, is indistinguishable from a "finger cloud". So we would expect my action (the observers) to be consistent with seeing a finger cloud - the idea beeing that I do not respond to information that is not in my possesion. This can be plausibly seen as a kind of "locality" - the local action depends on locally available information only.

Furthermore this means that if all other observers (say the entire environment) ALSO only sees just a finger cloud, the entire environment will be in consistency with you having a finger cloud.

If you ponder this, there is a difference between a random action, based on one of the possibilities, or one action based on ALL of the possibilities (the cloud). And this is what nature seems to be like.

Of course this view of rational action upon subjective information is just one possible interpretation.

/Fredrik

 

[Demystifier]

If you can't give me an example of a typical trajectory for an electron in a hydrogen atom, then I don't know what your theory is good for.

If the wave function has definite quantum numbers n,l,m then the trajectory is a circle with the angular velocity proportional to m. In particular, when m=0 (which is the case when n=0) then the particle is at rest.

 

[twofish-quant]

I'm thinking that there should be a list of "physics metaphors not to use" since it seems to me a lot of the problem is that people are explaining the Heisenberg uncertainty principle through an analogy with every day objects, and that ultimately leads to more confusion. (Explaining relativity as increasing mass should also be on that list.)

It may best to start with the statement that particle-waves are weird so don't expect that they behave like things we are use to. They aren't like anything that you encounter in the macroscopic world. The HUP is a fundamental property of that weirdness, and it's not the result of photons hitting electrons.

The other important thing is that the weirdness that you see is observational. Yes it may make more sense if we lived in a world where electrons acted like billiard balls, but they don't. Ultimately a lot of the questions of physics that start with "why?" end up with the answer "I don't know, but this is what I'm seeing."

 

[Demystifier]

They aren't like anything that you encounter in the macroscopic world.
...
Yes it may make more sense if we lived in a world where electrons acted like billiard balls, but they don't.
...
Ultimately a lot of the questions of physics that start with "why?" end up with the answer "I don't know, but this is what I'm seeing."

If you admit that you don't know what really happens at the microscopic level, then how can you be so sure about what does NOT happen?

 

[Longstreet]

It's a power law problem. Our brains had to evolve with a certain scale of objects where the wave nature is irrelevant to us in survival, at least until the last hundred years where we could see it. If we could have evolved in the quantum world it might seem weird that objects could retain precise states between measurements on larger scales then ourselves. Imagine if every time you walked in the room and your coffee mug was in the same place you left it, you would have to be paranoid who else had been in the room to make it stay there haha.

 

[RUTA]

It may best to start with the statement that particle-waves are weird so don't expect that they behave like things we are use to. They aren't like anything that you encounter in the macroscopic world.

Where's the cut between the microworld and macroworld?

 

[conway]

1s orbital. There, you have your answer.

Okay. But your answer is different from Demystifier's answer. Also, the 1s orbital is a charge cloud and I thought the Bohmian theory was supposed to give me an actual trajectory.

 

[conway]

If the wave function has definite quantum numbers n,l,m then the trajectory is a circle with the angular velocity proportional to m. In particular, when m=0 (which is the case when n=0) then the particle is at rest.

So in the ground state of the hydrogen atom, the electron just sits there hovering at a fixed distance from the proton?

 

[DrChinese]

Okay. But your answer is different from Demystifier's answer. Also, the 1s orbital is a charge cloud and I thought the Bohmian theory was supposed to give me an actual trajectory.

This is where things get complicated for the Bohmian, although you shouldn't expect Demystifier to agree. There are those who have attempted to work through the Bohmian trajectory (it can get complicated). I don't want to take this thread into the area of one interpretation or another, but you might gain some information from this:

http://arxiv.org/abs/0903.3878

No need to debate this paper itself, and here is the other side of the coin for the sake of balance:

http://arxiv.org/abs/0804.4564

 

[kanato]

If the wave function has definite quantum numbers n,l,m then the trajectory is a circle with the angular velocity proportional to m. In particular, when m=0 (which is the case when n=0) then the particle is at rest.

Uh.. how does that work? If the electron in a 1s state is at rest, then why does it have a nonzero expectation value for its kinetic energy? (By the virial theorem, the electron should have a kinetic energy expectation value of <p^2/2m> = +13.6 eV.) I can understand that the particle could be at rest but have uncertainty in position, but I don't understand how the particle can be at rest and have kinetic energy at the same time.

 

[Demystifier]

So in the ground state of the hydrogen atom, the electron just sits there hovering at a fixed distance from the proton?

Yes.

 

[Demystifier]

Uh.. how does that work? If the electron in a 1s state is at rest, then why does it have a nonzero expectation value for its kinetic energy? (By the virial theorem, the electron should have a kinetic energy expectation value of <p^2/2m> = +13.6 eV.) I can understand that the particle could be at rest but have uncertainty in position, but I don't understand how the particle can be at rest and have kinetic energy at the same time.

That is a good (and very frequent) question. When the electron is in the ground state, then its kinetic energy is zero. However, when you MEASURE the kinetic energy of the electron, then the electron becomes entangled with the measuring apparatus (which is also made up of quantum particles), so the electron is NO LONGER in the ground state. Instead the electron starts to move and attains a definite kinetic energy. It turns out, and this is THE CENTRAL part of Bohmian mechanics, that whatever you MEASURE, the probabilities of obtaining particular measurement outcomes are exactly the same as those given by standard QM.

The crucial point to remember is the fact that measurement changes the properties of the system. This fact is called - contextuality.

 

[Demystifier]

I don't want to take this thread into the area of one interpretation or another, but you might gain some information from this:

http://arxiv.org/abs/0903.3878

This paper (just as many similar ones) is wrong for a simple reason that it does not take into account the theory of quantum measurements involving entanglement between the measured system and the measuring apparatus. When this is taken into account, then, as Bohm proved GENERALLY in his 1952 paper, the Bohmian QM and standard QM have exactly the same predictions in all circumstances, as long as one measures observables defined by hermitian operators in the Hilbert space.

But for some reason, people tend to have an opinion on Bohmian mechanics without styding its crucial part - the theory of quantum measurements. Without that, Bohmian mechanics cannot be properly understood. It is the crucial part, even more important than particle trajectories themselves.