Does a particle really try every possible path?

[gerbilmore]

Hi,

I'm reading that a particle located at any given point can, in the next moment, be at any other given point in the universe. My understanding is that this is a correct interpretation of quantum physics.

So, my question is, how does the particle 'decide' where actually to be in the next moment, and why there rather than anywhere else? My current understanding is that it has something to do with the path of least action, which I've interpreted as it takes the 'easiest' route between the two points.

Assuming I'm correct so far, does this mean that the particle does NOT actually go to every point in the universe (even though it has to potential to do so), but actually only goes (or 'contemplates' going) to a relatively few points in the nearby area, which provides it with enough information to then 'know' which path is the 'easiest' to take?

My understanding is that this is similar to all the possible routes I could take on a trip from my house to the shop at the end of the street. I could get to the shop via Pluto or via a galaxy billions of light years away, but I don't, because I only have to contemplate walking a greater distance from the shop in the opposite direction before I give up on that path and go a shorter route, excluding my trip to Pluto from my plans. Would it be correct therefore to say that particles are efficient?

Am I totally confused? Thanks :-)

 

[Nugatory]

We have no way of knowing whether the particle actually tries every path. For that matter, it's not clear what it means to talk about a particle "trying" anything; it's not like a rat in a maze trying every path to get at the cheese.

It would be more accurate to say "You can calculate the probability of the particle being at any position by doing the right calculations with the every possible path and combining them in the right way".

 

[gerbilmore]

Thanks. So, would I be right in saying that the probability of a particle's next position being right next to its previous position has the highest probability, which is why particles don't jump around the universe from moment to moment but generally end up where we'd logically expect them to.

 

[Nugatory]

Thanks. So, would I be right in saying that the probability of a particle's next position being right next to its previous position has the highest probability, which is why particles don't jump around the universe from moment to moment but generally end up where we'd logically expect them to.

Pretty much, yes. In fact, if you calculate the "expectation value" of the position, which is what you'd get if you set up an arbitrarily large number of systems the same way, then measure the position of the particle for each one and take the average... You'll get the result that classical mechanics and what we'd logically expect of non-quantum particles predicts.

 

[scotian280]

So what happens when we bring it up to a double slit experiment and the particle becomes a wave, the a particle again? What do the calculations look like for that particle making it though the slits in some form or another and arriving on the other side?

 

[Nugatory]

So what happens when we bring it up to a double slit experiment and the particle becomes a wave, the a particle again?

That's not how the double-slit experiment works; the particle is never either a particle or a wave. Search this forum for some discussion of why "wave-particle duality" is misleading for more informatiuon.

The sum-over-all-paths approach produces the right answer when you include all the possible paths through both slits and include none of the paths that are blocked by the screen.

 

[gerbilmore]

Thanks all.

So, on this basis, would I be right in saying that something which at first glance appears quite mind-blowing i.e. that a particle—being more a 'cluster of probability' than a solid bit of stuff—can jump from one place to any other place in the universe in one unit of Planck time, is actually about as mind-blowing as me saying that I myself could be anywhere in the universe at the next moment from now. In other words, not at all—it really is still possible, but is just so vastly improbable that it may as well be described as impossible for all practical purposes. If this is the case, it seems that so much of the initial confusion and 'spookiness' of this aspect of QM disappears, to my mind at least!

 

[naima]

In the path integral are there ftl paths?

 

[bhobba]

So, on this basis, would I be right in saying that something which at first glance appears quite mind-blowing i.e. that a particle—being more a 'cluster of probability' than a solid bit of stuff—can jump from one place to any other place in the universe in one unit of Planck time,

One unit of plank time, particles jumping from one place to another - where are you getting this stuff from?

When not measured what a quantum particle is doing is anyone's guess.

Here is the detail of the path integral formalism

You start out with <x'|x> then you insert a ton of ∫|xi><xi|dxi = 1 in the middle to get ∫...∫<x|x1><x1|...|xn><xn|x> dx1...dxn. Now <xi|xi+1> = ci e^iSi so rearranging you get ∫...∫c1...cn e^ i∑Si.

Focus in on ∑Si. Define Li = Si/Δti, Δti is the time between the xi along the jagged path they trace out. ∑ Si = ∑Li Δti. As Δti goes to zero the reasonable physical assumption is made that Li is well behaved and goes over to a continuum so you get ∫L dt.

Now Si depends on xi and Δxi. But for a path Δxi depends on the velocity vi = Δxi/Δti so its very reasonable to assume when it goes to the continuum L is a function of x and the velocity v and you get the principle of least action because nearby paths generally cancel.

Strictly speaking its actually a hidden variable formulation - the path is the hidden variable - but of a very novel type.

Certainly particles are not jumping around the place in plank time.

Thanks
Bill

 

[bhobba]

In the path integral are there ftl paths?

No. Generally nearby paths cancel, but for stationary action they in fact reinforce, and since ftl paths can't exist, neither can the paths in the Feynman integral.

Thanks
Bill

 

[gerbilmore]

One unit of plank time, particles jumping from one place to another - where are you getting this stuff from?

I wasn't saying I was right, I was asking the question. So I don't mean jump I mean 'travel' ... essentially do whatever a particle does to get from A to B, where A is its starting point and B is anywhere else in the universe. And by one unit of Planck time I'm referring to the shortest possible time between two events. Maybe I'm mistaken, but this is my understanding of things. I'm currently reading this book:

http://en.wikipedia.org/wiki/The_Quantum_Universe

Sorry, but everything else you've written makes no sense to me right now!

 

[phinds]

I wasn't saying I was right, I was asking the question. So I don't mean jump I mean 'travel' ... essentially do whatever a particle does to get from A to B, where A is its starting point and B is anywhere else in the universe. And by one unit of Planck time I'm referring to the shortest possible time between two events. Maybe I'm mistaken, but this is my understanding of things. I'm currently reading this book:

http://en.wikipedia.org/wiki/The_Quantum_Universe

Sorry, but everything else you've written makes no sense to me right now!

Your understanding of how fast a particle could move from one place to another would allow faster than light travel. This is impossible. If someone told you that quantum objects travel from any point to any other point in one Plank Time they were either kidding you or making things up. More likely you misunderstood what was said/written. Some quantum objects, the photon for example, travel AT the speed of light but none travel faster and massive ones, the electron for example, travel slower.

 

[gerbilmore]

More likely you misunderstood what was said/written.

Seems to be the case, hence my questioning :)

So, let me revisit my question:

With reference to the double slit experiment, would I be right in saying that something which at first glance appears quite mind-blowing i.e. the suggestion that a particle (as a wave) takes every possible route from the source to the screen, is as mind-blowing as me saying that I too could take every possible route from my house to the local shop (some of which may go via Pluto or another galaxy were I to travel at the speed of light). In other words, not at all mind-blowing—yes, it really is still possible according to the laws of physics, but most physically possible journeys between my house and the local shop are just so vastly improbable that they may as well be described as impossible for all practical purposes. If this is the case, it seems that so much of the initial confusion and 'spookiness' of this aspect of QM disappears, to my mind at least!

Can you see what I'm getting at?

 

[bhobba]

That book is a popularisation. Brian Cox tries hard to explain QM, but without math you always run into problems. I have a copy and he makes a reasonable fist of it - but perfect it aren't - or rather in trying to express these concepts without math issues will always arise. QM does not say everything that can happen does happen. Feynman's parth integral formulation does not say particles literally takes all possible paths - it says they mathematically behave like they take all possible paths.

I strongly suggest you study Susskinds book instead:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

It requires some math but it is correct.

There are also associated video lectures:
http://theoreticalminimum.com/

When you have gone through that book you should be able to understand what I posted.

None of the generally accepted theories of physics require a shortest possible unit of time - its continuous.

Thanks
Bill

 

[gerbilmore]

Thanks. I will certainly look into the book and watch the videos.

As an aside...

None of the generally accepted theories of physics require a shortest possible unit of time - its continuous.

From: http://simple.wikipedia.org/wiki/Planck_time

"Theoretically, this is the shortest time measurement that is possible."

?

 

[bhobba]

Can you see what I'm getting at?

No.

You seem to be reading things into QM it doesn't say. That way only leads to confusion.

Please study Susskinds book. It requires a smattering of calculus but the following suffices:
https://www.amazon.com/dp/0471827223/?tag=pfamazon01-20

Also reading his first book will help as well:
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20

Thanks
Bil

 

[bhobba]

Theoretically, this is the shortest time measurement that is possible."

That article shows a common misunderstanding of the uncertainly principle. It does not put a limit on measurement precision - rather its a statistical statement about measurements of similarly prepared systems.

In our most powerful and experimentally verified theory, Quantum Field Theory, time is continuous.

Thanks
Bill

 

[gerbilmore]

Thanks. I recognise there subject matter is highly complex, but there's so much contradictory information out there; it's hard to know what to believe! The videos look like a great immediate starting point though - much appreciated.

 

[phinds]

Seems to be the case, hence my questioning :)

So, let me revisit my question:

With reference to the double slit experiment, would I be right in saying that something which at first glance appears quite mind-blowing i.e. the suggestion that a particle (as a wave) takes every possible route from the source to the screen, is as mind-blowing as me saying that I too could take every possible route from my house to the local shop (some of which may go via Pluto or another galaxy were I to travel at the speed of light). In other words, not at all mind-blowing—yes, it really is still possible according to the laws of physics, but most physically possible journeys between my house and the local shop are just so vastly improbable that they may as well be described as impossible for all practical purposes. If this is the case, it seems that so much of the initial confusion and 'spookiness' of this aspect of QM disappears, to my mind at least!

Can you see what I'm getting at?

I agree w/ all of bhobba's statements but I think I do see what you are getting at and yes, other than your mis-statement about your being able to travel at the speed of light, you are right. These weird things are possible (but one at a time, not all together ... it's a statistical thing) but almost all paths are so utterly improbably that they can be ignored for all practical purposes. I'm not sure that changes the weirdness of QM but it's also clear that you have a distorted view of QM so it may well change what YOU view as the weirdness of QM.

 

[gerbilmore]

Thanks. I'll get there! :)

other than your mis-statement about your being able to travel at the speed of light

Would you mind explaining what you mean? Why a misstatement? I could, in theory, travel at the speed of light, couldn't I?

it's also clear that you have a distorted view of QM so it may well change what YOU view as the weirdness of QM

That's true. My understanding of QM began, many years ago, quite by accident, with http://en.wikipedia.org/wiki/What_the_Bleep_Do_We_Know!? (hey, we all have to start somewhere!). I now realize that simplifying things can strip a great deal of meaning and lead to all sorts of wrong thinking. I'm aware that QM is weird, but you're right, it's perhaps not as weird as I first thought.

 

[phinds]

Would you mind explaining what you mean? Why a misstatement? I could, in theory, travel at the speed of light, couldn't I?

No, you most emphatically could not. Nothing with mass can travel at c.

 

[naima]

No. Generally nearby paths cancel, but for stationary action they in fact reinforce, and since ftl paths can't exist, neither can the paths in the Feynman integral.

The propagator gives the amplitude x(t')|x(t)
Does Feynman say that we can avoid paths going forward and backward in time and get the correct result?

 

[bhobba]

The propagator gives the amplitude x(t')|x(t) Does Feynman say that we can avoid paths going forward and backward in time and get the correct result?

Did you see post 9?

Why do you think it would allow particles traveling back in time or FTL?

Are you trying to handle amti-particles? That was Feynmans particular way of looking at it - I don't think its an inherent part of the formalism.

Thanks
Bill

 

[stevendaryl]

Did you see post 9?

Why do you think it would allow particles traveling back in time or FTL?

Well, your post #9 didn't say anything about restricting the integral over $x_i$ so that $|\frac{x_i - x_{i-1}}{\delta t_i}| < c$

 

[StevieTNZ]

A good article that I found just now: http://phys.org/news/2015-01-atoms.html?utm_source=menu&utm_medium=link&utm_campaign=item-menu

"We have now used indirect measurements to determine the final position of the atom in the most gentle way possible," says the PhD student Carsten Robens. Even such an indirect measurement (see figure) significantly modified the result of the experiments. This observation excludes – falsifies, as Karl Popper would say more precisely – the possibility that Caesium atoms follow a macro-realistic theory. Instead, the experimental findings of the Bonn team fit well with an interpretation based on superposition states that get destroyed when the indirect measurement occurs. All that we can do is to accept that the atom has indeed taken different paths at the same time.

 

[bohm2]

A good article that I found just now: http://phys.org/news/2015-01-atoms.html?utm_source=menu&utm_medium=link&utm_campaign=item-menu

"Instead, the experimental findings of the Bonn team fit well with an interpretation based on superposition states that get destroyed when the indirect measurement occurs. All that we can do is to accept that the atom has indeed taken different paths at the same time."

Note that there are certain limitations to the conclusions the authors draw in the paper cited:

Yet, the concept of well-defined trajectories in position space can, in part, still be rescued, provided that one renounces locality. An example is provided by Bohmian mechanics, whose predictions are shown to be equivalent to those of nonrelativistic quantum mechanics. In this interpretation of quantum theory, physical objects follow precise trajectories, which are guided by the Universe’s pilot wave function, that is, by a physical entity constituting a nonlocal hidden variable. It is therefore clear that Bohmian mechanics is not in contradiction with our findings since, from that point of view, assumption (A2) is not fulfilled.

Ideal Negative Measurements in Quantum Walks Disprove Theories Based on Classical Trajectories
http://journals.aps.org/prx/pdf/10.1103/PhysRevX.5.011003

 

[Ookke]

"All that we can do is to accept that the atom has indeed taken different paths at the same time."

If that is the case, shouldn't the mass of the particle be multiplied by a huge factor? If it's at multiple places at the same time. Or should we interpret that only fractions of the particle take different paths, and the combined mass of the fractions is that of the particle?

 

[StevieTNZ]

If that is the case, shouldn't the mass of the particle be multiplied by a huge factor? If it's at multiple places at the same time. Or should we interpret that only fractions of the particle take different paths, and the combined mass of the fractions is that of the particle?

No - the system doesn't exist in the classical sense. It exists as a potentiality (unless Bohmian Mechanics is correct).

 

[bhobba]

Well, your post #9 didn't say anything about restricting the integral over $x_i$ so that $|\frac{x_i - x_{i-1}}{\delta t_i}| < c$

No - but its obvious unless <x|x'> is physically resizeable its going to be zero - and since particles, relativistically can't travel FTL or go backward in time that's not possible.

Th caveat here is the Feynman-Stueckelberg interpretation of antiparticles as particles traveling backwards in time. In that view you would include FTL paths - but they aren't really traveling FTL or going backward in time - its simply an elegant way of handling the math. This is seen by the fact a particle can be considered an ani-particle traveling backward in time due to the symmetry of the situation. Its simply an elegant way to interpret it.

Thanks
Bill

 

[Ookke]

No - the system doesn't exist in the classical sense. It exists as a potentiality (unless Bohmian Mechanics is correct).

Ok. Maybe it's then misleading to talk about "paths" at all. However if we let a test particle travel through a room so that it doesn't interact until the back wall, it's tempting to think that something travels through the room, and the mass, energy and momentum are conserved in some form, even though it's not a particle in the classical sense.

 

[bhobba]

If it's at multiple places at the same time

Its been said many many times before but it seems a particularly stubborn idea to try and view quantum systems as doing something when not observed - being in many places at once, taking many parths etc etc. QM is a theory about observations. When not observed the theory is silent. Its not in two places at once, taking multiple paths. As far as we can tell its not doing anything.

The sum over histories approach is simply saying mathematically its like the particle takes many paths simultaneously - it not saying that's what's actually going on. Strictly speaking its a hidden variable interpretation but of a very novel type - the path is the hidden variable - the novelty is it doesn't take one particular path - but all of them.

Thanks
Bill

 

[Ookke]

As far as we can tell its not doing anything.

#30 could be my reply to your post as well. Maybe it's completely irrelevant in physics what is happening between interactions, but it's interesting to think.

 

[phinds]

Ok. Maybe it's then misleading to talk about "paths" at all. However if we let a test particle travel through a room so that it doesn't interact until the back wall, it's tempting to think that something travels through the room, and the mass, energy and momentum are conserved in some form, even though it's not a particle in the classical sense.

Yeah, I think that way too. My logic is that if you move the wall a bit closer, you still get an interaction. Move it a bit closer and you still get an interaction. And so forth. So clearly it's THERE in some sense.

The mistake would be to connect the dots between all those interactions and think that you have found even one path that the particle took on the way to the wall when it was farthest away. There's no path, there's nothing until you get an interaction, but I'm agreeing w/ you that it is in some sense there even if that is undefined and not useful in practice.

 

[bhobba]

#30 could be my reply to your post as well. Maybe it's completely irrelevant in physics what is happening between interactions, but it's interesting to think.

You are looking at it incorrectly. QM is not saying its irrelevant what's happening between observations - its silent about it.

The question is why do you want to add things to a theory that doesn't say anything about it and discuss that with people who have gotten used to this perplexing part of QM? All you are going to get is what I have said - its not part of the theory.

Thanks
Bill

 

[stevendaryl]

No - but its obvious unless <x|x'> is physically resizeable its going to be zero - and since particles, relativistically can't travel FTL or go backward in time that's not possible.

Th caveat here is the Feynman-Stueckelberg interpretation of antiparticles as particles traveling backwards in time. In that view you would include FTL paths - but they aren't really traveling FTL or going backward in time - its simply an elegant way of handling the math. This is seen by the fact a particle can be considered an ani-particle traveling backward in time due to the symmetry of the situation. Its simply an elegant way to interpret it.

Thanks
Bill

A related question on this topic: If you do a path integral using the nonrelativistic lagrangian $L = \frac{1}{2} m v^2 - V$, you get the nonrelativistic propagator
$G(x, t, x, t) = \langle x| e^{\frac{-i}{\hbar} H t} | x'\rangle$

If instead you do a path-integral with a relativistic Lagrangian such as: $L = - m \sqrt{g_{\mu \nu} U^\mu U^\nu}$ or $L = \frac{1}{2} m U^\mu U^\nu$, does that give the Klein-Gordon propagator?