How can a particle have infinite positions in quantum mechanics?

[StevieTNZ]

Last night I started reading the section of Brian Cox's "The Quantum Universe" where they discuss how to calculate the probability of a particle being found at a particular position.

What he has stated first is if we have an initial position, at any later time we can find the particle at any spot in the universe. Okay, all good so far.

An example is given where we roughly know where the particle is initially, and we want to calculate the probability of it being at spot X in the future. All possible routes from where it could be initially are calculated producing a final probability for X. Now they're saying because of interference, most of the routes cancel each other out (at least, I get that impression). But in the end, the conclusion is it "effectively has no chance of being found at X".

Effectively no chance? No probability at all, or a small probability? In one breath you say a particle can hop to every other position in the universe in an instance (even if its in a superposition, saying it can be anywhere would seem to me that it does NOT have probability 0 of being at particular points. When you say something is in a superposition of state A and state B and state C, you wouldn't say its in a superposition of all three if state C had no chance of occurring).

Would it be correct from the get-go we need to consider the particle to be in every possible position in the universe to calculate the probability we find it at position X?

In all honesty reading this section has made me more confused.

 

[bhobba]

What I think you are talking about is the sum over histories approach and why most paths cancel. The answer I would give depends on how much math you know but since I can't really explain it without a bit of math I will give the answer with the least amount.

Each path is represented by a complex number Ce^if - f is called the action. Both C and f depend on position and time. You may remember f can be interpreted as an angle and if it is rotated by 180 degrees then the number is the negative of what it was before. Now if f is a very large number then if f actually varies depending on the path a very close path can be found such that f changes by 180 degrees and since it is the negative of the other path they cancel. This will always happen with one exception - if for that close path f did not change - in which case instead of cancelling it will add. The paths where f does not change are called the paths of least action and, for all practical purposes, are the only ones that exist. This explains a very important principle of classical physics called the principle of least action and that principle alone, believe it or not is all that is required to derive all of classical mechanics.

Check out:
http://www.eftaylor.com/leastaction.html
http://www.eftaylor.com/pub/QMtoNewtonsLaws.pdf

If you are interested in the technicalities of how Classical Mechanics is derived this way the book to get is Landau Mechanics:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

'If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages. I first came across Landau's mechanics many years ago as a brash undergrad. My prof at the time had given me this book but warned me that it's the kind of book that ages like wine. I've read this book several times since and I have found that indeed, each time is more rewarding than the last.'

Many people feel like that after reading the book - it had a profound effect on me and from what I can gather many others that have read it.

Thanks
Bill

 

[Maui]

What I think you are talking about is the sum over histories approach and why most paths cancel. The answer I would give depends on how much math you know but since I can't really explain it without a bit of math I will give the answer with the least amount. Bill

Math is not reality, it's just a tool that for some reason is successful in mimicking reality to a certain level of accuracy. WRT that, the above explanation is neither successful in addressing the question(as posed) nor complete. AFAICS, the question is more geared towards the interpretation side of things.

 

[Maui]

What he has stated first is if we have an initial position, at any later time we can find the particle at any spot in the universe. Okay, all good so far.

The inability to know a position of a free particle anywhere in the universe follows from the wave nature of 'particles' and the HUP as well(this cannot be interpreted, as you well know, and is taken to be fundamental). And if it's really fundamental(we have been wrong before), it says a lot about realism.

 

[bhobba]

Math is not reality, it's just a tool that for some reason is successful in mimicking reality to a certain level of accuracy. WRT that, the above explanation is neither successful in addressing the question(as posed) nor complete. AFAICS, the question is more geared towards the interpretation side of things.

Of course math is not 'reality' - it however can be, and in this case is, a model of 'reality' (whatever that is) with strong experimental support. Next thing you will be claiming is the phasor diagrams engineers use to describe electric circuits can't possibly work because its just math.

Now precisely what don't you understand about what I wrote? Not a philosophical objection but the nitty gritty and nuts and bolts of it?

Thanks
Bill

 

[Maui]

Of course math is not 'reality' - it however can be, and in this case is, a model of 'reality' (whatever that is) with strong experimental support. Next thing you will be claiming is the phasor diagrams engineers use to describe electric circuits can't possibly work because its just math.

No, you fail to understand that things in reality don't work because the math says so(math has been adjusted to fit observations as much as possible and as close as possible, not the other way around). It would be a misconception and certainly not a mainstream view to claim otherwise. Moreover, you failed to demonstrate how the 'interpretation' of phasor diagrams is representative or relevant to interpreting the formalism of quantum mechanics and why you think such an analogy would be useful.

Now precisely what don't you understand about what I wrote? Not a philosophical objection but the nitty gritty and nuts and bolts of it?

Thanks
Bill

Do 'particles' follow each path that is represented by a complex number as the 'nuts and bolts' of mathematics imply considering that you claimed of mathematics - "in this case is, a model of 'reality'" You know this? Really?

You said "

The paths where f does not change are called the paths of least action and, for all practical purposes, are the only ones that exist

whereas the OP asked if it is possible in practice to

Now they're saying because of interference, most of the routes cancel each other out (at least, I get that impression). But in the end, the conclusion is it "effectively has no chance of being found at X".

Comparing the bolded text, do you still find your answer satisfactory?

 

[bhobba]

No, you fail to understand that things in reality don't work because the math says so(math has been adjusted to fit observations as much as possible and as close as possible, not the other way around)?

No one I am aware of claims that - I certainly do not. The claim is models (and QM is a model) that have been found to be correct by experiment to a high degree of accuracy have logical deductions that that have the same level of confidence as the experiments that support the model. It's the same thing that for example allows us to predict how a spinning top behaves from the mathematics of classical mechanics.

Phasors were not mentioned as having anything to do with QM - it was mentioned as another example of a mathematical model that is used extensively and no one doubts its applicability.

I think you need to think more about what a mathematical model is:
http://www.math.uAlberta.ca/~devries/erc2001/slides.pdf

Comparing the bolded text, do you still find your answer satisfactory?

Yes indeed. You do understand in QM states are modeled by complex numbers and that key fact is the reason for interference and that most paths are canceled by a very close path? This is combined with the fact that the f I mentioned usually has a factor 1/hbar in front of it I omitted for simplicity and since hbar is very small it makes it very large. Have you read Feynmans QED - The Strange Theory Of Light And Matter? The turning arrow he talks about is a complex number which has both length and an angle and explains what I said in a slightly different way. If you haven't here is a link to the actual lectures:
http://vega.org.uk/video/subseries/8

Thanks
Bill

 

[questionpost]

An example is given where we roughly know where the particle is initially, and we want to calculate the probability of it being at spot X in the future.

Nope, HUP says that since we measure it and have completely determined its position, its momentum is completely undetermined and thus it could exist at any energy level. We can't base the information of the future on the present. All we can do is gather data and look for patterns, such as that "we keep seeing it appear in this general area, and it has a few specific energies , so we can say that when energy=x1, x2, x3, it will generally appear over in these regions".

All possible routes from where it could be initially are calculated producing a final probability for X. Now they're saying because of interference, most of the routes cancel each other out (at least, I get that impression). But in the end, the conclusion is it "effectively has no chance of being found at X".

"routs" don't really cancel each other out in just a plain standing wave, what happens is just the position itself is undetermined, kind of like how "1/0" is undefined or doesn't have a specific outcome since it can technically yield any number as a result.

Effectively no chance? No probability at all, or a small probability? In one breath you say a particle can hop to every other position in the universe in an instance (even if its in a superposition, saying it can be anywhere would seem to me that it does NOT have probability 0 of being at particular points. When you say something is in a superposition of state A and state B and state C, you wouldn't say its in a superposition of all three if state C had no chance of occurring).

At large distances probability comes so close to zero that scientists just consider it 0, just like how the mass of an electron in chemistry is so small we don't consider it. We still only have general areas of probability of measuring a particle at a given energy even if it can appear somewhere else in the universe. Matter isn't completely a particle or completely a wave, you could sort of say that it exists more where its more probable and exists less where its less probable.

 

[jewbinson]

Effectively no chance? No probability at all, or a small probability?

In all honesty reading this section has made me more confused.

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

What you're referring to is "almost never", which is a similar concept to "almost surely".

Edit: oh yeah, I forgot to mention: an event with probability 0 CAN happen. For example, in the darts scenario on the unit square (the example provided by wiki), the probability of hitting the point (1/2,1/2) = 0, BUT it is still POSSIBLE!

 

[Maui]

No one I am aware of claims that - I certainly do not. The claim is models (and QM is a model) that have been found to be correct by experiment to a high degree of accuracy have logical deductions that that have the same level of confidence as the experiments that support the model. It's the same thing that for example allows us to predict how a spinning top behaves from the mathematics of classical mechanics.

Wouldn't it have been much easier for you to supply a straightforward answer to the question as posed in the OP, instead of beating about the bush:

But in the end, the conclusion is it "effectively has no chance of being found at X".

Since you believe the formalism is true because it's empirically verified, the ONLY obvious answer is that according to the SE the answer is that there is a non-zero chance that the particle can be at X,Y,Z at time T somewhere in the universe, whereby specifying that the time T may come after 2 billion years or more(or less).



How is the following quote from the OP not a philosophical question?(did you read the question at all?)

Would it be correct from the get-go we need to consider the particle to be in every possible position in the universe to calculate the probability we find it at position X?

You do understand in QM states are modeled by complex numbers and that key fact is the reason for interference

Absolutely NOT. You are again confusing models for reality and deluding yourself that everything flows out of the formalism, whereas the only thing that can be said categorically is that interference is CAUSED by the wave nature of 'particles'. I don't find this point particularly hard to grasp. I would guess the reason the comedy arises in the first place is that an instrumentalist attempts to answer an otherwise obvious philosophical question and tries to sweep the philosophical implications under the rag with comments like - for all practical purposes. You could have simply said - qm is not classical mechanics and things don't behave the same way as they tend to do macroscopically.

 

[bhobba]

Wouldn't it have been much easier for you to supply a straightforward answer to the question as posed in the OP, instead of beating about the bush:

Wouldn't it be easier for you to make an attempt to understand it instead of confusing your inability to do so with the belief I did not supply an answer?

You do understand in QM states are modeled by complex numbers and that key fact is the reason for interference and that most paths are canceled by a very close path?

Absolutely NOT.

Figures. How about actually reading even a basic textbook?

Thanks
Bill

 

[Maui]

Wouldn't it be easier for you to make an attempt to understand it instead of confusing your inability to do so with the belief I did not supply an answer?

What you supplied was not an answer to the question but to a hypothetical question that was not(asked about practical purposes). Why is that not fairly obvious by now?

Figures. How about actually reading even a basic textbook?

Thanks
Bill

Setting aside the adhominem attack, my point was that the probability of finding a particle at any point (its "probability density") was related to the square of the height of the probability wave at that point. That quickly vanishing amplitude extends throughout space(with non-zero values) which answers the opening post.

 

[bhobba]

What you supplied was not an answer to the question but to a hypothetical question that was not(asked about practical purposes). Why is that not fairly obvious by now?

Maybe because the confusion lies with you rather than what I wrote?

Setting aside the adhominem attack, my point was that the probability of finding a particle at any point (its "probability density") was related to the square of the height of the probability wave at that point. That quickly vanishing amplitude extends throughout space(with non-zero values) which answers the opening post.

What kind of number is a probability amplitude in QM?

I was going to leave it at that but to try and stop this going on longer than necessary here is the answer:
http://en.wikipedia.org/wiki/Probability_amplitude
'In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density. For example, if the probability amplitude of a quantum state is α, the probability of measuring that state is | α | 2. The values taken by a normalised wave function ψ at each point x are probability amplitudes, since |ψ(x)|2 gives the probability density at position x.'

It is the fact it is a complex number that is responsible for interference effects and for nearby paths canceling - exactly as my original post explained. Now I again ask what about do you not understand? Is it because you don't know what a complex number is? No shame in that. If you don't then have a look at the Feynman lectures I gave a link to - he explains it without using that concept.

Maybe if I explain it another way it will be clearer. The probability of finding a particle at any point is the square of a complex number called the probability amplitude. In the sum over history approach that number is the contribution of the sum over all possible paths ie you need to add up all the complex numbers that each path gives at that point. Now consider the contribution of some path - it turns out to have the form Ce^if (which is of course the form of any complex number) where f has the form of g/hbar - the fact hbar is small is very important. Since hbar is very small this means f will be a very large number. Now consider a close path such that f changes by pi. Since this is such a large number C will not change much for such a close path so that you end up with minus Ce^if, which when added to the original path results in a big fat zero - ie the contribution of that path is cancelled. There is only one case where this will not happen - where f does not change when a small change in the path is made - in which case it will add instead of cancel. This means only the paths where f does not vary over the path are left as contributing to the probability of finding a particle - and it usually turns out to be just one path where that happens. And that is why in most cases the particle follows a distinct path - it is the path where when you make a small change in the path the probability amplitude does not change.

Thanks
Bill

 

[questionpost]

If you graph the function of a probability wave, there is a horizontal asymtote at x=0 which means the probability never reaches 0 but only comes infinitely close to it.

 

[StevieTNZ]

Please please. No need to have a go at each other. I have found each and every post helpful. Thank you all for the answers. :)

 

[StevieTNZ]

So to clarify:

Every quantum system in the universe, evolving according to Schrodinger's equation, has a non-zero probability of being anywhere else in the universe at a later time? Would that even include a particle being used in a lab experiment?

 

[questionpost]

So to clarify:

Every quantum system in the universe, evolving according to Schrodinger's equation, has a non-zero probability of being anywhere else in the universe at a later time? Would that even include a particle being used in a lab experiment?

You can't predict where a particle will show up based on present information. Or I guess you can say that you can, but even in a very isolated system you still have like a 50% chance of being wrong.
If you look at some graphs of probability waves, there are some very specific places where the wave intersects the x-axis, but after those x values it just asymtotes. Of course as time goes on, that probability changes, so the places that use to intersect the x-axis are no longer there and there is never a consistant place that the probability is 0.

 

[StevieTNZ]

I'm not sure if that answers my question. Of course we cannot predict where a particle will be at a later time with certainty. We can, however, predict probabilities for every possible result from every future possible experiment conceivable based on a starting wave function.

 

[questionpost]

I'm not sure if that answers my question. Of course we cannot predict where a particle will be at a later time with certainty. We can, however, predict probabilities for every possible result from every future possible experiment conceivable based on a starting wave function.

I don't think we can do that either, unless Douglas Adams invented the infinite improbability drive in his slot in the cemetery.

 

[StevieTNZ]

From point A to point B, a particle can take an infinite amount of paths to get to the next point. What paths are possible? Ones that don't currently have other systems in them?

 

[questionpost]

From point A to point B, a particle can take an infinite amount of paths to get to the next point. What paths are possible? Ones that don't currently have other systems in them?

Particles don't "take paths" because if they did they would radiate their energy away. Instead, they merely have an undefined vector state, which is different than physically taking paths.

 

[StevieTNZ]

When you calculate the probability of a particle landing on a screen behind double slits, do we take into account all the paths possible from every point in the universe (would the particle evolved to spread everywhere before reaching the slits?) Of course, would it would need to be histories that involve the experiment, rather than a particle at point C (say where the screen is in another history) to point D (another point on the screen in another history) without there being the experiment happening? Or would you include every possible history that allows the particle to land at the point on the screen, even if the history doesn't involve a double slit experiment?

 

[lugita15]

When you calculate the probability of a particle landing on a screen behind double slits, do we take into account all the paths possible from every point in the universe (would the particle evolved to spread everywhere before reaching the slits?) Of course, would it would need to be histories that involve the experiment, rather than a particle at point C (say where the screen is in another history) to point D (another point on the screen in another history) without there being the experiment happening? Or would you include every possible history that allows the particle to land at the point on the screen, even if the history doesn't involve a double slit experiment?

In principle you have to consider every possible history in which the initial and final configurations of your system are the same, however exotic they may be. You can have the particle traveling a little bit, taking a brief detour to the Andromeda galaxy, the screen going to Mars, and then the particle and the screen both returning at exactly the right time so that the particle hits the screen. However, the contributions due to these exotic histories tend to cancel each other out to an extremely good approimation, so that you'll usually get excellent results if you just stick to more realistic paths.

 

[StevieTNZ]

In principle you have to consider every possible history in which the initial and final configurations of your system are the same, however exotic they may be. You can have the particle traveling a little bit, taking a brief detour to the Andromeda galaxy, the screen going to Mars, and then the particle and the screen both returning at exactly the right time so that the particle hits the screen. However, the contributions due to these exotic histories tend to cancel each other out to an extremely good approimation, so that you'll usually get excellent results if you just stick to more realistic paths.

What I mean is the initial position of the particle. Do we calculate the paths from every point in the universe, to the point on the screen?

 

[lugita15]

What I mean is the initial position of the particle. Do we calculate the paths from every point in the universe, to the point on the screen?

No, to calculate probabilities in quantum mechanics we need to specify both the initial condition and the final condition. A general question you answer in QM is, "Given a particle in the state state |ψ>, what is the probability that it will be measured in the state |χ>?" In this case, the specific question is, "Given a particle with position x0 at time t0, what is the probability that it will be measured with position at at time t?" To do that, you add up the amplitude for each history in which the particle started out at (x0,t0) and finished at (x,t), regardless of how exotic they are. Here (x0,t0) is the time and place the particle was launched, and (x,t) is the time and place it's detected at the screen.

 

[StevieTNZ]

I guess the basis of my question was based on that the Schrodinger equation has evolved a lot and when it predicts a particle after t=0 is everywhere at once, you have everything everywhere.

 

[lugita15]

I guess the basis of my question was based on that the Schrodinger equation has evolved a lot and when it predicts a particle after t=0 is everywhere at once, you have everything everywhere.

Yes, you have everything having a probability of being anywhere. But of course, they can have more probability of being in certain regions rather than other regions. Your head is much more likely to stay attached to your body than to disappear and reappear on Venus.

 

[StevieTNZ]

Yes, you have everything having a probability of being anywhere. But of course, they can have more probability of being in certain regions rather than other regions. Your head is much more likely to stay attached to your body than to disappear and reappear on Venus.

Thank goodness for whoever set those probabilities into the laws of physics!

So I gather: you calculate paths from points to the same place (point B) if the experimental set-up is the same in each of those histories? (rather than calculate getting to point B without there being a double slit experiment)

 

[lugita15]

Thank goodness for whoever set those probabilities into the laws of physics!

So I gather: you calculate paths from points to the same place (point B) if the experimental set-up is the same in each of those histories? (rather than calculate getting to point B without there being a double slit experiment)

Both point A, the initial point, and point B, the final point, must be the same in all histories your summing over. And more generally, the initial configuration of your whole system and the final configuration of your whole system must be the same in all the histories you're summing over. This includes the photon source, the slits, the screen, etc.

 

[StevieTNZ]

Gotcha.