Understanding the Uncertainty Principle: Momentum & Position

[neophysique]

I'm trying to understand the Uncertainty Principle,
for the case of momentum and position, but I'm stumped.
Suppose one can measure the position of a particle
with arbitrary precision. Suppose the measurement
of this particle's positions were made at time t= 0 and
then time t=1. The vector length of the particle path
divided by time then would yield the velocity of
the particle at time t=1, per definition. If the mass
of the particle was also constant then per definition
mv= momentum should yield the exact momentum
of the particle at t=1 , no?

If the above is accurate, how does one get to an
Uncertainty Principle? I'm sure the math leads
to this Principle but I'm having a hard time
relating it physically.

 

[Rach3]

I'm trying to understand the Uncertainty Principle,
for the case of momentum and position, but I'm stumped.
Suppose one can measure the position of a particle
with arbitrary precision. Suppose the measurement
of this particle's positions were made at time t= 0 and
then time t=1. The vector length of the particle path
divided by time then would yield the velocity of
the particle at time t=1, per definition.

This leads to inconsistencies. For example, you'd expect the position at t=1/2 to be halfway between the position at t=0 and t=1, classicaly, right? Do all three measurements; this isn't the case. Did something accelerate? The more position measurements you do, the closer together you space them in time... the more wildly they flucutate and disagree with each other. Even in an extremely short time interval, they might be observed at two very remote positions, "implying" extreme velocity. Or they might stay in the same spot.

Of course each position measurement was arbitraily precise - except then the momentum measurements you're implicity performing get a large statistical distribution.

(This explanation was inspired by the very first chapter of Landau & Lif****z, authors whose names are inexplicably censored by this forum's software. You can actually read that part for free in the excerpt on Amazon's page (link)).

 

[Rach3]

If the above is accurate, how does one get to an
Uncertainty Principle? I'm sure the math leads
to this Principle but I'm having a hard time
relating it physically.

It comes from the formalism of Quantum Mechanics, which has its own postulates and cannot be derived from classical physics since it actively disagrees with classical physics.

 

[neophysique]

This leads to inconsistencies. For example, you'd expect the position at t=1/2 to be halfway between the position at t=0 and t=1, classicaly, right? Do all three measurements; this isn't the case. Did something accelerate? The more position measurements you do, the closer together you space them in time... the more wildly they flucutate and disagree with each other. Even in an extremely short time interval, they might be observed at two very remote positions, "implying" extreme velocity. Or they might stay in the same spot.

Of course each position measurement was arbitraily precise - except then the momentum measurements you're implicity performing get a large statistical distribution.

(This explanation was inspired by the very first chapter of Landau & Lif****z, authors whose names are inexplicably censored by this forum's software. You can actually read that part for free in the excerpt on Amazon's page (link)).

Perhaps the results would depend on the type of experiments
performed. The results you describe seemed to come from
experiments in which the particle's starting and ending points
are not confined in space.

The forum software censors that name probably cause it contains
a curse word.

 

[HallsofIvy]

How do you determine the position? Imagine you have a really large magnifying glass so that you can actually see the electron. You still have to shine a light on it so that you can see it. That light carries momentum, furthermore to get a very accurate position you will need to use small wavelength light (you can only get the postion to within a half wavelength of the light you use) which has higher momentum. Although you can use the two positions to accurately calculate the speed (and momentum) of the electron before you shown the light on it, the light itself changes the momentum.

 

[DrChinese]

Perhaps the results would depend on the type of experiments performed.

The HUP is actually independent of the type of experiment being performed. There is no question that you can observe the non-commuting attributes of a particle. The question is: what do you learn from such observations? If you cannot use the results of those observations to predict values at some future point in time, you didn't really learn as much as you thought. After measuring position and momentum, you will always find inconsistent results with follow-on measurements. This is true of any non-commuting observables. For example, measuring the y spin component of an electron completely invalidates any previous x or z spin component measurement, but NOT subsequent y spin measurements (or other observables that do commute). That is the HUP.

 

[Gokul43201]

I'm trying to understand the Uncertainty Principle,
for the case of momentum and position, but I'm stumped.
Suppose one can measure the position of a particle
with arbitrary precision. Suppose the measurement
of this particle's positions were made at time t= 0 and
then time t=1. The vector length of the particle path
divided by time then would yield the velocity of
the particle at time t=1, per definition.

No, that's not right. You have only determined an average velocity during the interval.

 

[RandallB]

If the mass
of the particle was also constant then per definition
mv= momentum should yield the exact momentum
of the particle at t=1 , no?

Simplifying is good to help understand, but you’re glossing over some significant points in your OP example. Restating the problem let your test particle follow a single straight line and we can give you both a start time of t=0 and a distance of d=0 precisely. Your measurements at “About” d=10 away at t=1 need to produce precise measurements relative to that starting coordinate.

Here’s the problem
1) You can actually measure the WHERE the particle is by detecting its exact crossing of the d=10.000 distance exactly --- BUT the more exact the location measurement is the less precise (uncertain) is your measurement of WHEN the particle arrived there! E.G. the time comes out between 0.99? to 1.00?
2) BUT wait you can revise your measurement to see exactly WHEN it arrives to see it as precisely t=1.000; BUT now this new measurement is unclear as to exactly WHERE it is with results like d = 9.99? to 10.00? so now you know WHEN but are uncertain about WHERE.

So as you put it nailing down the WHEN would give you the momentum your looking for to go with your WHERE measurement. The trick is getting both WHEN and WHERE at the same time.

Now here is the part your need to get, this not because of HUP quantum uncertainty -- these are experimental observations made I the 1920’s, and EXPLAINED by HUP uncertainty.
What was claimed by that explanation – that HUP could predict the amount of that uncertainty and no measurement could ever be better than that prediction.

But hold on, you’ve got better rulers and better stop watches than they ever had 80 years ago, and two hands to use both at once, surly you can now do better than they did back then.
NOPE – HUP so far has predicted correctly; so well in fact that the theoretical principles associated with that are largely responsible for most of the high technology created in the 20th century.

Exactly why is it this way? – plenty of competing theories working on being the one that explains that in detail, -- including some that claim there is no real improvement to be made over what we already know – no winner yet.

 

[Rach3]

Simplifying is good to help understand, but you’re glossing over some significant points in your OP example. Restating the problem let your test particle follow a single straight line and we can give you both a start time of t=0 and a distance of d=0 precisely. Your measurements at “About” d=10 away at t=1 need to produce precise measurements relative to that starting coordinate.

Here’s the problem
1) You can actually measure the WHERE the particle is by detecting its exact crossing of the d=10.000 distance exactly --- BUT the more exact the location measurement is the less precise (uncertain) is your measurement of WHEN the particle arrived there! E.G. the time comes out between 0.99? to 1.00?
2) BUT wait you can revise your measurement to see exactly WHEN it arrives to see it as precisely t=1.000; BUT now this new measurement is unclear as to exactly WHERE it is with results like d = 9.99? to 10.00? so now you know WHEN but are uncertain about WHERE.

So as you put it nailing down the WHEN would give you the momentum your looking for to go with your WHERE measurement. The trick is getting both WHEN and WHERE at the same time.

Now here is the part your need to get, this not because of HUP quantum uncertainty -- these are experimental observations made I the 1920’s, and EXPLAINED by HUP uncertainty.
What was claimed by that explanation – that HUP could predict the amount of that uncertainty and no measurement could ever be better than that prediction.

This is quite wrong. There's no non-commutivity between position operators X and time t, which isn't even an operator but a parameter. HUP places no restriction on simultaenous measurements of position and time; even if some alien formalism had such a thing, it wouldn't have units of action (hbar) but rather of distance*time = distance^2, so it would clearly be incompatible with the uncertainty principle of QM.

 

[Eye_in_the_Sky]

I'm trying to understand the Uncertainty Principle,
for the case of momentum and position, but I'm stumped.
Suppose one can measure the position of a particle
with arbitrary precision. Suppose the measurement
of this particle's positions were made at time t= 0 and
then time t=1. The vector length of the particle path
divided by time then would yield the velocity of
the particle at time t=1, per definition. If the mass
of the particle was also constant then per definition
mv= momentum should yield the exact momentum
of the particle at t=1 , no?

If the above is accurate, how does one get to an
Uncertainty Principle? I'm sure the math leads
to this Principle but I'm having a hard time
relating it physically.

The Uncertainty Principle is about "what will be", not "what was".

Or, if you want to fix your eye only on the past, it is about "what could have been", not just "what was".

 

[RandallB]

This is quite wrong. ...incompatible with the uncertainty principle of QM.

Nonsense: If QM allowed that at one observation you could measure the exact location and time at that location; and also be able to measure with precision both time and location at second observation how would you not be able to define a precise Momentum and Location for the particle?
It is the essence of the OP ? by neophysique.
He deserves a response that addresses the context of his question; not some theory detail like position operators in units of hbar, or some philosophy like – what is, will be, what was, to what will be.

 

[Rach3]

The Uncertainty Principle is about "what will be", not "what was".

Or, if you want to fix your eye only on the past, it is about "what could have been", not just "what was".

You might want to be more specific, some of us do not follow you.

 

[Rach3]

Nonsense: If QM allowed that at one observation you could measure the exact location and time at that location; and also be able to measure with precision both time and location at second observation how would you not be able to define a precise Momentum and Location for the particle?

No. There's nothing in HUP against measuring both position and momentum, simulatenously, to arbitrary precision - that's a misconception. The uncertainty is not the "measurement uncerainty", the precision of the measuring apparatus, it is a statistical uncertainty over ensembles of identical systems. Read my above explanation or the linked excerpt from the opening of Landau's book, for an illuminating example.

Better yet, fall back on the formalism. What happens to a wavefunction when you measure it's position? How does it evolve after that? And how much does a later position measurement tell you?

Remember that position and time aren't on equal footing in non-relativistic QM formalism. Time is a parameter, which is always known to arbitrary precision.

 

[Eye_in_the_Sky]

You might want to be more specific, some of us do not follow you.

Suppose that at time t₁ the system is found to be at position x₁ with spread equal to ∆x. More precisely, at time t₁ we perform a filtering-type measurement for which the output wavefunction is some φ(x) centered at x₁ with standard deviation ∆x.

The Uncertainty Principle then tells us:

Any subsequent measurement of position will give us a result x₂ which lies in some interval of x-values consistent with a momentum uncertainty on the order of h/∆x.

This refers to "what will be".

Specifically, suppose that at time t₂ the system is found to be at position x₂. The Uncertainty Principle merely informs us that the value of x₂ is quite likely to fall in the above mentioned associated interval.

Of course, having made that measurement, we can then infer a momentum for the system over the duration of the time interval (t₁,t₂). This refers to "what was".

Next, if we then look back on the whole affair, we realize that the result x₂ which we obtained did not have to be what is was. Rather, as required by the Uncertainty Principle, it could have been anything consistent with the associated interval of x-values. And this, of course, refers to "what could have been".

I repeat:

The Uncertainty Principle is about "what will be", not "what was".

Or, if you want to fix your eye only on the past, it is about "what could have been", not just "what was".

Is what I meant clear enough now?

 

[RandallB]

Better yet, fall back on the formalism.

Better than that, fall back on why they needed to create that “formalism” and the view used by QM.
QM gives a better solution to describing WHEN and WHERE (location & momentum) at atomic level measurements, than classical. Making it more useful to follow the QM explanation, even as it turned out to be classically non-local.

You don’t have start with the chicken and demand they understand QM theory, you can start with the egg and how a solution to real problems was raised from that egg.

 

[prochatz]

How do you determine the position? Imagine you have a really large magnifying glass so that you can actually see the electron. You still have to shine a light on it so that you can see it.

I don't think that the problem is the method we use. The Uncertainty Principle does not depend on the method. It's a part of our world. The wrong must be somewhere else. (I totally agree with the procedure and the results you mentioned).

In my opinion, neophysique' s thought is wrong on the derivation. I don't even know if we can use the length vector in such problems. It seems too classical.

 

[masudr]

...then per definition mv= momentum...

In quantum mechanics, observables are not functions of co-ordinate space.

The transition from CM to QM is easily made from the Hamiltonian formulation. In this approach, our state is defined on a state space, separate from the phase space.

Observables are operators defined on the state space, where the form of the operator comes from replacing x by the $\hat{x}$ operator and p by the $\hat{p}.$ In the x-basis, these become $\hat{x}=x$ and $\hat{p}=-i\hbar d/dx$.

Therefore, per definition, possible values of momentum is given by the eigenvalues of the momentum operator, not mv.

 

[Eye_in_the_Sky]

In my opinion, neophysique' s thought is wrong on the derivation. I don't even know if we can use the length vector in such problems. It seems too classical.

The prescription offered by neophysique is a valid one. It is correct to say that at time t=1 the particle is at the place in which it has been found to be, and moreover, that it must have had a particular momentum in order to get there.

Here are Feynman's words on this sort of question:

It is quite true that we can receive a particle, and on reception determine what its position is and what its momentum would have had to have been to have gotten there. That is true, but that is not what the uncertainty relation (2.3) refers to. Equation (2.3) refers to the predictability of a situation, not remarks about the past.

["The Feynman Lectures on Physics", vol. 3, p. 2-3]

 

[Eye_in_the_Sky]

I must clarify

The prescription offered by neophysique is a valid one. It is correct to say that at time t=1 the particle is at the place in which it has been found to be, and moreover, that it must have had a particular momentum in order to get there.

This requires clarification.

The said "validity" and "correctness" is understood to be in a context where 'permission' is granted to engage in a semi-classical type of analysis. Beginner and intermediate textbooks on Quantum Mechanics are replete with examples of such analyses, the purpose of which is to give the reader a deepened understanding, albeit from an informal, and perhaps it is even fair to say "ill-defined", point of view.

So, strictly speaking (i.e. when 'permission' is not granted) there is no room for neophysique's analysis. It is as prochatz says, the "thought is wrong on the derivation ... It seems too classical", and as masudr explains, "possible values of momentum is given by the eigenvalues of the momentum operator, not" m(x₂-x₁)/∆t.

Nonetheless, if we accept that 'permission' has been granted and we want to push our classical way of thinking as far as it can go, then a rule like ∆x∙∆p~h (now stripped away from its purely formal -- but unambiguous -- meaning in terms of operators and state vectors) needs to be clarified. What I have indicated in my earlier posts, that the Uncertainty Principle must then apply only to prediction, but not to retrodiction, gets the job done.

The above having been said, it is now a good idea to go back and look at the problem again, this time around from a strictly "orthodox" perspective. So, here we go:

A measurement of position is performed at time t=0, and then again at time t=1. During the intermediate times 0<t<1, we cannot say that the particle must have been somewhere, and that therefore it must have described some kind of path. No, on the contrary, all we can say is that over the course of those intermediate times the particle was in a state of "superposition of possibilities" described by a wavefunction evolving according to Schrödinger's equation, and then, at time t=1, a particular possibility became actualized.

If we ask the question, "What path did the particle take to get from its initial position to its final position?", then we must answer that the particle had some amplitude to go along each and every path, and when we compute these amplitudes and combine these paths according to the appropriate rules, then we are able to compute the probability for the particle to reach its final position, given its initial position. In short, since we did not measure the path, the particle did not have a (particular) path. Therefore, the particle did not have a well-defined momentum over the course of the intermediate times 0<t<1.

That having been said, perhaps now someone else would like to explain it all over again, this time around from the perspective of MWI, and then again, from the perspective of Bohm.

 

[HallsofIvy]

I don't think that the problem is the method we use. The Uncertainty Principle does not depend on the method. It's a part of our world. The wrong must be somewhere else. (I totally agree with the procedure and the results you mentioned).

In my opinion, neophysique' s thought is wrong on the derivation. I don't even know if we can use the length vector in such problems. It seems too classical.

My point was not that it depended on the particular method of observation but rather to show how, for a particular method, the observation necessarily changes the situation. Some other method of observation would entail a different mechanism but still change the situation. By "accurately" observing the position of a particle twice you can calculate the momentum just before the second observation but that will not be the momentum after the observation.

 

[Rach3]

If we ask the question, "What path did the particle take to get from its initial position to its final position?", then we must answer that the particle had some amplitude to go along each and every path, and when we compute these amplitudes and combine these paths according to the appropriate rules, then we are able to compute the probability for the particle to reach its final position, given its initial position. In short, since we did not measure the path, the particle did not have a (particular) path. Therefore, the particle did not have a well-defined momentum over the course of the intermediate times 0<t<1.

But we can still make very important statistical statemenents! For example, things like
$$\langle \psi(t=0.5) | \hat{P} | \psi(t=0.5) \rangle$$ and
$$\langle \psi(t=0.5) | \hat{P}^2 | \psi(t=0.5) \rangle$$

which tell us the likely statistics of momentum measurements at an intermediate time t=0.5 in our ensemble. And there's a correspondence with classical physics, because we find that our semiclassical (x2-x1)/(t2-t1) is actually the quantum expectation value of P/m - which means the average over all possible momentum outcomes would be the classical value!

 

[Eye_in_the_Sky]

Yes, Rach, I most definitely agree!

 

[neophysique]

How do you determine the position? Imagine you have a really large magnifying glass so that you can actually see the electron. You still have to shine a light on it so that you can see it. That light carries momentum, furthermore to get a very accurate position you will need to use small wavelength light (you can only get the postion to within a half wavelength of the light you use) which has higher momentum. Although you can use the two positions to accurately calculate the speed (and momentum) of the electron before you shown the light on it, the light itself changes the momentum.

I'm thinking of those "slit"-type experiments. Both the slit
and the screen are at a fixed distance from another in time.
If clocks and detectors were placed at the slit and the screen,
and one particle at a time was shot from the source then
we would know the positions of the particle at two points
in time, in this reference frame, with precision. Using this
velocity, one can calculate the exact momentum of the particle
at the slit after it interacted with the detector, yes?

Sure, everytime the particle hits a detector, its momentum changes.
But we can set up a row of detector/screens and merely focus
on two positions at a time to calculate the exact momentum of
the particle from the previous screen, et al.

 

[Rach3]

I'm thinking of those "slit"-type experiments. Both the slit
and the screen are at a fixed distance from another in time.
If clocks and detectors were placed at the slit and the screen,
and one particle at a time was shot from the source then
we would know the positions of the particle at two points
in time, in this reference frame, with precision. Using this
velocity, one can calculate the exact momentum of the particle
at the slit after it interacted with the detector, yes?

Measuring the position of the particle at the double-slit will kill the interference effects.

 

[neophysique]

Measuring the position of the particle at the double-slit will kill the interference effects.

Yes, but the particle will still strike the screen, leaving a history
of when and where it struck the screen, no? We are deriving
the momentum of one particle so the cumulative effect
of multiple particles on the screen is irrelevant to the calculation.