Does electron has definite path?

[Godparicle]

Let us suppose that, at definite time intervals Δt, successive measurements of the coordinates of an electron are made. The results will not in general lie on a smooth curve. On the contrary, the more accurately the measurements are made, the more discontinuous and disorderly will be the variation of their results, in accordance with the non-existence of a path of the electron . A fairly smooth path is obtained only if the coordinates of the electron are measured with a low degree of accuracy, as for instance from the condensation of vapour droplets in a Wilson chamber.
If now, leaving the accuracy of the measurements unchanged, we diminish the intervals Δt between measurements, then adjacent measurements, of course, give neighbouring values of the coordinates. However, the results of a series of successive measurements, though they lie in a small region of space, will be distributed in this region in a wholly irregular manner, lying on no smooth curve. In particular, as Δt tends to zero, the results of a adjacent measurements by no means tend to lie on one straight line.
This circumstance shows that, in quantum mechanics, there is no such concept as the velocity of a particle in the classical sense of the world, i.e. the limit to which the difference of the coordinates at two instants, divided by the interval Δt between these instants, tends a Δt tends to zero. However, we shall see later that in quantum mechanics, nevertheless, a reasonable definition of the velocity of a particle at a given instant can be constructed, and this velocity at any given instant, in quantum mechanics the situation is entirely different. If, as a result of measurement, the electron is found to have definite coordinates, then it has no definite velocity whatever. Conversely, if the electron has a definite velocity, it cannot have a definite position in space.For the simultaneous existence of the coordinates and velocity would mean the existence of a definite path, which the electron has not. Thus, in quantum mechanics, the coordinates and velocity of an electron are quantities which cannot be simultaneously measured exactly, i.e they cannot simultaneously have definite values...

If we diminish the intervals Δt between measurements to zero, we can definitely know the path of the electron. Isn't it?

 

[Drakkith]

You can't reduce the time interval between measurements to zero, otherwise the measurements are taking place at the same time. Such a situation is not a realistic one.

 

[Godparicle]

Such a situation is not a realistic one.

Why?

 

[Drakkith]

Why?

Because measurements take a finite amount of time. You can't have the interval between them be zero. You can't even do the math with the interval set to zero.

 

[atyy]

The question is tricky. It seems that (the theory allows) you to reduce the time between measurements so that one gets at every time a definite position reading, and hence a path. However, this path is random, and the position measurement is not an accurate measurement of the position observable. I don't know if this is the only possibility for a continuous measurement, and additional details are given in http://arxiv.org/abs/quant-ph/0611067.

Regarding Landau and Lifshitz's statement that an electron in non-relativistic quantum mechanics cannot have a trajectory, we have to say it with the following qualifications for the statement to be true: an electron cannot have a definite trajectory in which it has at all times simultaneously well defined position and momentum, where position and momentum are the quantum mechanical canonically conjugate position and momentum which reduce to the classical canonically conjugate position and momentum. The mathematics behind this statement is that the Wigner function of an electron is in general not a probability distribution. Another relevant bit of mathematics is the Kochen-Specker theorem.

If we remove the restriction, we can assign the electron a trajectory. In fact, there are many different ways consistent with quantum mechanics of assigning an electron a deterministic trajectory with random initial conditions. These correspond to the large variety of possible Bohmian interpretations.

 

[Godparicle]

Because measurements take a finite amount of time. You can't have the interval between them be zero.

Even if I agree with you that interval b/w measurements to be not zero. Electron does has path, isn't it? It doesn't matter whether we measure it or not. Otherwise, it would mean that electron dematerialised in one position and recreated in another position, it would become science fiction-teleportation, which I don't think is the case with electron here.

You can't even do the math with the interval set to zero.

I am sorry, I am not aware of this. Can you be more explicit on this?

 

[bhobba]

Can you be more explicit on this?

Its the limit thing of calculus.

You can take the limit as time goes to zero and get Feynman's sum over histories - but it actually can't be zero.

BTW at a rigorous level this approach has a lot of mathematical difficulties - but that is another (deep) story requiring some hairy stuff from white noise theory.

Also note the sum over histories approach is actually a hidden variable theory but of a rather non-trivial type. The path is the hidden variable.

Thanks
Bill

 

[bhobba]

Electron does has path, isn't it? It doesn't matter whether we measure it or not.

What properties it has when not observed the theory is silent about - and that includes its path.

QM is a theory about observations here in a common-sense classical world.

The real issue with QM is how does it explain such a world that it assumes in the first place - but that is another issue. If you are interested start a new thread.

Thanks
Bill

 

[carllooper]

The concept of an electron traveling a well defined path is one possible way of explaining/describing an electron detection - for one can propose that an electron detection is the result of an electron following a well defined path to the point of detection.

But an electron path is not, in itself, an observable. It is a concept. And to that extent we can say it exists. It exists as a concept. Or an explanation (or a description), for what is otherwise an observable, such as an electron detection. To the extent that the concept of a well defined path explains, or describes, or elaborates an electron detection, it will be considered a good concept. To the extent that it doesn't it will be considered not such a good concept.

It so happens that the concept of an electron following a well defined path has failed to explain (or describe, or elaborate) the observation of interference patterns (amongst other things) in electron detections, so other better concepts have been sought. And these have required giving up (to some extent, if not entirely) the concept of an electron following a well defined path.

It's not that anyone ever wanted to give up such a concept. Because obviously the concept of a well defined path would have been a lot easier to apply to an observable. Rather it is observables (such as interference patterns) that have inspired (or required) alternatives.

Indeed it is observables that inspired the concept of well defined path in the first place. But on closer inspection the observables have inspired/required alternative (if related) concepts such as superposition, or Feynman diagrams, and so on.

Concepts are not descriptions or explanations of some mysterious hidden unobservable universe. They are ways of explaining, or understanding, or describing, or elaborating, in terms of language (such as mathematics amongst others) what is otherwise observable. Not what isn't.

C

 

[DrChinese]

The concept of an electron traveling a well defined path is one possible way of explaining/describing an electron detection - for one can propose that an electron detection is the result of an electron following a well defined path to the point of detection.

But an electron path is not, in itself, an observable. It is a concept. And to that extent we can say it exists. It exists as a concept. Or an explanation (or a description), for what is otherwise an observable, such as an electron detection. To the extent that the concept of a well defined path explains, or describes, or elaborates an electron detection, it will be considered a good concept. To the extent that it doesn't it will be considered not such a good concept.

It so happens that the concept of an electron following a well defined path has failed to explain (or describe, or elaborate) the observation of interference patterns (amongst other things) in electron detections, so other better concepts have been sought. And these have required giving up (to some extent, if not entirely) the concept of an electron following a well defined path.

It's not that anyone ever wanted to give up such a concept. Because obviously the concept of a well defined path would have been a lot easier to apply to an observable. Rather it is observables (such as interference patterns) that have inspired (or required) alternatives.

Indeed it is observables that inspired the concept of well defined path in the first place. But on closer inspection the observables have inspired/required alternative (if related) concepts such as superposition, or Feynman diagrams, and so on.

Concepts are not descriptions or explanations of some mysterious hidden unobservable universe. They are ways of explaining, or understanding, or describing, or elaborating, in terms of language (such as mathematics amongst others) what is otherwise observable. Not what isn't.

C

Sorry, I don't see your point. As a USEFUL concept: I see that its utility lies in having something to talk about in day to day casual conversation. But if you take it too far, it becomes an interpretation-dependent idea. Bohmians believe particles have definite trajectories. Most others don't. There have been plenty of papers written discussing this in great detail. Here's one to consider:

http://arxiv.org/abs/0903.3878

"The Bohmian interpretation of quantum mechanics adds particle trajectories to the wave function and ensures that the probability distribution of the particle positions agrees with quantum mechanics at any time. This is not sufficient to avoid contradictions with quantum mechanics. There are correlations between particle positions at different times which cannot be reproduced with real particle trajectories. A simple rearrangement of an experimental test of the Bell-CHSH inequality demonstrates this."

So if you go too far with the concept, it will probably be simply wrong.

 

[Khashishi]

There is no way to know exactly when you made a measurement. If you have two closely timed measurements, you can't always tell which one came first, so how can you assign a trajectory?

 

[.Scott]

In particular, as Δt tends to zero, the results of a adjacent measurements by no means tend to lie on one straight line.

Of course, as Δt tends to zero, the amount of energy applied to the electron to measure it over any set time period tends to infinity.

 

[Nugatory]

Regarding Landau and Lifshitz's statement that an electron in non-relativistic quantum mechanics cannot have a trajectory, we have to say it with the following qualifications for the statement to be true: an electron cannot have a definite trajectory in which it has at all times simultaneously well defined position and momentum, where position and momentum are the quantum mechanical canonically conjugate position and momentum which reduce to the classical canonically conjugate position and momentum.

In other words, an electron may have something that we might call a "trajectory", but that something does not very closely resemble what a layman (or a classical physicist in a pragmatic mood) means by the word.

 

[carllooper]

Sorry, I don't see your point. As a USEFUL concept: I see that its utility lies in having something to talk about in day to day casual conversation. But if you take it too far, it becomes an interpretation-dependent idea.

Yes, the concept of a well defined path can be useful in day to day casual conversation. That's a good point. And yes, if you take it too far, it can become quite an interpretation-dependent idea. I'd agree with that as well. Indeed, I'd suggest one doesn't have to take the concept very far at all before it becomes interpretation-dependent.

Re. the point I was making - if you fail to see it, perhaps that's because you haven't actually read it. My point, if you did or do bother to actually read it, is not in any way agreeing with the proposition of a well defined path.

Indeed I make it quite obvious that such a concept has been abandoned (to the extent it has). But more importantly, I make quite obvious why it has been abandoned. It is because it doesn't fulfil what we might otherwise require of concepts in science: that they explain, or describe, or elaborate what is otherwise observable.

The layman is at a disadvantage because they may not fully appreciate either the observables in question and/or the particular language (such as mathematics) in which such observables are elaborated.

They will inherit (or even suppose off their own bat) ancient concepts such as well defined paths, and not, it might be said, because they are in any way incapable of adopting newer concepts.

And then there are those who are just quite happy to sit within a given concept, ignoring what they see. Or could see, if they made some effort to do so.

C

 

[dextercioby]

Leaving diff,geom. finesse aside, a classical path of a pointlike particle in dynamics (Lagrangian or Hamiltonian) is a smooth mapping from the real axis (time) to the 3-dim configurations space. There's nothing probabilistic about it. In quantum mechanics, a quantum path is essentially a probabilistic notion*: in Heisenberg picture it's a smooth assignment from the real axis to the set of all expectation values of the 3 coordinate operators assuming a known state. The time evolution of these expectation values -which would correspond to the Lagrange equations- is governed by the so-called Ehrenfest theorem which was rigorously proven only in 2009 - G. Friesecke and B. Schmidt, “A sharp version of Ehrenfest’s theorem for general self-adjoint operators,”

*The probabilistic nature of quantum mechanical mathematical objects is expressed by the so-called Born rule.

 

[atyy]

Leaving diff,geom. finesse aside, a classical path of a pointlike particle in dynamics (Lagrangian or Hamiltonian) is a smooth mapping from the real axis (time) to the 3-dim configurations space. There's nothing probabilistic about it. In quantum mechanics, a quantum path is essentially a probabilistic notion*: in Heisenberg picture it's a smooth assignment from the real axis to the set of all expectation values of the 3 coordinate operators assuming a known state. The time evolution of these expectation values -which would correspond to the Lagrange equations- is governed by the so-called Ehrenfest theorem which was rigorously proven only in 2009 - G. Friesecke and B. Schmidt, “A sharp version of Ehrenfest’s theorem for general self-adjoint operators,”

*The probabilistic nature of quantum mechanical mathematical objects is expressed by the so-called Born rule.

It looks like this is a different concept of a quantum path than what one obtains by continuous measurement?

In Friesecke and Schmidt's paper http://rspa.royalsocietypublishing.org/content/466/2119/2137.full it looks like one measures sharp position only once on each particle, but the measurement is carried out at a different time for every particle in the ensemble. In contrast, in the stochastic Schroedinger equation in http://arxiv.org/abs/quant-ph/0611067 one measures unsharp position continuously on the same particle?

 

[dextercioby]

I didn't address the particular issue of a continuous measurement, the only connection to the <physicality> is the one through the quoted article, which is actually a piece of applied functional analysis. The <physicality> is indeed given by an ensemble and a single sharp position measurement at the same time for all systems in the ensemble, then let time pass and repeat the measurement; that's how you end up with a time function of expectation values assuming the states are <forced> not to evolve, which is of course the piece of unphysical assumption of the Heisenberg picture.

 

[bhobba]

There is no way to know exactly when you made a measurement. If you have two closely timed measurements, you can't always tell which one came first, so how can you assign a trajectory?

Cant follow that one - generally you can tell the ordering of measurements.

The issue is it takes time to measure position - but one can conceptually imagine taking a limit.

Thanks
Bill

 

[carllooper]

There are two types of observables in an interference experiment. One is a particle detection which is an immediate observable, in which a singular position and timestamp can be unambiguously assigned to that detection. It doesn't take place over time and over space. It takes place in time and in space. Conceptually we can argue the measurement destroys the particle. It's no longer measureable at a latter time. Indeed it's not even measureable at an earlier time. The other observable is the distribution of detections. Unlike a single detection, this distribution is not describable in terms of a single location, at an instant in time. It is a somewhat different concept that the observable provokes.

The relationship between these two observables, and the otherwise different concepts they might (or do) provoke, is that which further conceptualisation attempts to elaborate to the extent it can. For whatever purpose.

C

 

[bhobba]

There are two types of observables in an interference experiment.

There is only one - the position observable at the screen. But outcomes are registered at different times.

Thanks
Bill

 

[carllooper]

There is only one - the position observable at the screen. But outcomes are registered at different times.

That's like saying the only thing observable in an image of a house are the individual pixels. But this is not true. There is also observable the house.

In an interference experiment there is clearly observable a pattern in the distribution of the detections. An interference pattern: a variation in the density of the detections, across space, and across time.

This pattern is not invisible.

C

 

[Godparicle]

It seems that (the theory allows) you to reduce the time between measurements so that one gets at every time a definite position reading, and hence a path. However, this path is random, and the position measurement is not an accurate measurement of the position observable. I don't know if this is the only possibility for a continuous measurement, and additional details are given in http://arxiv.org/abs/quant-ph/0611067.

The following passage has been extracted from the link atyy has given:

"When measurement is first introduced to students of quantum mechanics, it is invariably treated by ignoring any consideration of the time the measurement takes: the measurement just "happens", for all intents and purposes, instantaneously. This treatment is good for a first introduction, but is not sufficient to describe two important situations. The first is when some aspect of a system is continually monitored.This happens, for example, when one illuminates an object and continually detects the reflected light in order to track the objects motion. In this case, information is obtained about the object at a finite rate, and one needs to understand what happens to the object while the measurement takes place. It is the subject of continuous quantum measurement that describes such a measurement. The second situation arises because of nothing really happens instantaneously. Even rapid, "single shot" measurements take some time. If this time is not short compared to the dynamics of the measured system, then it is once again important to understand both the dynamics of the measured system, then it is once again important to understand both the dynamics of the flow of information to the observer and the effect of the measurement on the system"

Light travels at the speed which no other can travel. The dynamics (speed) of the system or single electron (which we have considered) won't be greater than the speed of light, i.e we can track the objects motion by illuminating the object continually i.e, having zero time interval between the measurements. And thus we can have definite path of the electron.

 

[carllooper]

Observation, or measurement if one prefers, is conceived in terms of interactions between systems. It is understood that the interaction alters what would otherwise occur had the measurement/observation (interaction) not been made

By way of example, if we put our hand between a light and a screen, we can say the hand measures the light in the sense that it might, for example, appreciate a change in temperature. But the hand can also be understood as stopping light from going any further than the hand. One can appreciate this concept in the form of the shadow being cast on the screen.

Without some concept of the hand playing a role we might very well conclude from measuring light in this way, that an intrinsic property of light is to sometimes produce a shadow on the screen, in the shape of a hand. And at other times not.

And there's nothing necessarily wrong with such a concept. But we can elaborate the concept to include the role played by the hand. And in this way we can suggest the presence/absence of a shadow in the shape of a hand, need not be treated as some intrinsic property of light.

This can be useful if nothing else. For example, designing a video projector might be a bit awkward if one had to work with the concept that light might, every now and then, manifest shadows on the screen in the shape of hands. Better to assume that hands play a role in that.

C

 

[bhobba]

In an interference experiment there is clearly observable a pattern in the distribution of the detections. An interference pattern: a variation in the density of the detections, across space, and across time.

Sure - but that in no way changes each dot is a position observation, and each observation is exactly the same as all the others - just with, maybe, a different result. The wave-function gives the probability of each observation.

Thanks
Bill

 

[bhobba]

The dynamics (speed) of the system or single electron (which we have considered) won't be greater than the speed of light, i.e we can track the objects motion by illuminating the object continually i.e, having zero time interval between the measurements. And thus we can have definite path of the electron.

Can you describe, in detail, the apparatus that can be used to illuminate a single electron and track its motion?

QED says you can't do that (eg it may absorb the photon or scatter it), but if you have figured out how to overturn the most exactly verified physical theory of all time I am all ears.

Thanks
Bill

 

[carllooper]

Sure - but that in no way changes each dot is a position observation, and each observation is exactly the same as all the others - just with, maybe, a different result. The wave-function gives the probability of each observation.

Where do I suggest that each particle detection (a dot) is not describable in terms of a position?

It is the distribution (the pattern) of such detections (equally observable) that doesn't lend itself to being described in such a way.

One can however, for each particle detection, associate the concept of a wave function.

This allows one to compose a description of the distribution (the interference pattern) from the set of all wave functions, associated with each particle detection.

C

 

[bhobba]

Where do I suggest that each particle detection (a dot) is not describable in terms of a position?

That's not my issue - it is:

There are two types of observables in an interference experiment.

There is only one - the screen is a position observation - there is no other.

It is the distribution (the pattern) of such detections (equally observable) that doesn't lend itself to being described in such a way.

I suspect you don't quite understand what an observation in QM is.

By definition an observable is a Hermitian operator whose eigenvalues are the possible outcomes. Observations are the possible outcomes of observables. A pattern is not an observation.

Thanks
Bill

 

[carllooper]

I suspect you don't quite understand what an observation in QM is.

While QM is science, science is not just QM. QM emerges within science. The interference pattern constitutes a valid scientific observation in the sense that it is a clearly visible part of an interference experiment. But more to the point the observation, in this case, was and remains quite clearly defined as the distribution of particle detections across space and time.

The purpose of this definition is quite simple. It is to ensure it is not conflated with a particle detection.

To reconflate it with a particle detection serves no scientific purpose.

C

 

[Nugatory]

By definition an observable is a Hermitian operator whose eigenvalues are the possible outcomes. Observations are the possible outcomes of observables. A pattern is not an observation.

The interference pattern constitutes a valid scientific observation in the sense that it is a clearly visible part of an interference experiment. But more to the point the observation, in this case, was and remains quite clearly defined as the distribution of particle detections across space and time.

An observed distribution is pretty much by definition an emergent phenomenon... but then again, as far as QM is concerned, the entire macroscopic world is an emergent phenomenon.

Bhobba is of course right about the definition of "observable" in the mathematical formalism.

And if I'm understanding carllooper properly, he's using the word to also cover phenomena that, like the interference pattern, emerge from individual quantum "observations" (in the strict sense) in the same way that gas pressure emerges from individual molecules striking the walls of a vessel.

So if you guys are disagreeing about more than the appropriateness of carllooper's use of the word "observe", I don't see it. And if I'm mistaken and there is a deeper disagreement, I still don't see much connection to the topic of this thread. Am I missing something here?

 

[carllooper]

An observed distribution is pretty much by definition an emergent phenomenon... but then again, as far as QM is concerned, the entire macroscopic world is an emergent phenomenon.

There's no real disagreement. Not yet anyway.

I understand the friction. The term "observable" will have a particular meaning in the context of QM. And one might easily get a bit irritated with the use of that same term in relation to something such as an interference pattern - so let's just say: "for want of a better term". Even if there is no better term.

The point being made is that an interference pattern is not invisible. It is observable (for want of a better term). And it is in relation to the visibility of an interference pattern we will propose the concept of individual particles, conceived in terms of wave functions, and the deeper details of which will constitute the mathematical domain of QM. To put it another way: if there were no such thing as an interference pattern on the one hand, and particle detections on the other, the concepts of QM would not be the way they are.

To the extent that QM attempts to treat such observations, such as an interference pattern, as an "emergent phenomena", it will be consistent with the more general concept of the macroscopic, as a function of the microscopic (or the nanoscopic, subatomic, etc). This concept (of the large as a function of the small) is not an invention of QM. It finds it's roots in classical physics. Indeed it is a very old concept. Ancient one might say. But it is a particularly useful concept, despite it's age, especially in terms of technology. It lends itself to a very economic organisation of the physical world. One can encapsulate a whole range of phenomena in terms of a few simple concepts.

With respect to the question at hand: "do electrons have a definite path", the answer to be elaborated is that such is not a question. Not really. A particle having a definite path is firstly a concept. An ancient concept. But a concept nevertheless. And as such it is itself already an answer - just not necessarily a very good one. QM provides better concepts than the concept of a particle possessing a definite path.

As to why QM provides better concepts one can return to the observables (for want of a better term). The observables (for want of a better term) provoke the creation of better concepts.

C