# Doubt regarding uncertainty principle

I had a thought going in my head today about the uncertainty principle. Forgive me if it sounds too silly.

Consider an electron in motion. Now, suppose that I'm measuring it's position with infinite accuracy. So, by the uncertainty principle, it's velocity is blurry. But what if I measure the position of the electron again, after a very short interval of time, with infinite accuracy? Then, I can compare the distance it has moved from its previous position, and as I know the time taken for the electron to move this distance, i can figure out its velocity. So, I can measure both the position and velocity of the electron simultaneously, thereby violating the uncertainty principle.

So, what is the mistake I've made in this theoretical experiment?

martinbn
Velocity in quantum mechanics is something else. It is not the same as in classical mechanics.

Fredrik
Staff Emeritus
Gold Member
[strike]** Editing ** I misread your question a bit...give me a minute to fix.[/strike]

Edit: Sorry...I got distracted for a while, and my brain was a bit mushy after writing a long reply in another thread. I see that you got a few good answers in the mean time. I would say that the flaw in the argument you used is that it's circular. If you don't assume that both the position and velocity are well-defined between the measurements, you won't be able to conclude that you can find the velocity by measuring distance and time.

When you make the first position measurement (which we assume is extremely accurate), you're making the velocity extremely ill-defined. This means that the particle doesn't have a well-defined velocity to be "found out".

However, I think it should also be mentioned that if the particle starts out in a state of approximately well-defined position and approximately well-defined momentum, and both position measurements are less accurate than the position uncertainty, the measurements won't change the state by much, and you can determine an approximate velocity in the way you described.

Last edited:
alxm
Now make a third measurement. Will that value be consistent with your previous measurements? The answer here is that it won't be. And the more accurately your first two position measurements are made, the less consistent your third one will be.

If you measure the position of a particle multiple times with "infinite" frequency, you will obtain the same result each time, because you "collapse" the wavefunction to a dirac delta upon measurement of the position of the particle. However, if you wait sufficiently long, you will obtain a second measurement. However, if you do this multiple times, you measurements will not be consistent.

The uncertaintly principle is a statement about the statistical spread of measurements all performed on the same state. When you measure a system, you change it's state.

How to you measure the velocity of an electron, You pass a photon through it and measure the frequency change to know the probability density. Well, that action alone induces energy and increase in energy leads to increase in momentum ...then it becomes a probable position..

Fredrik
Staff Emeritus
Gold Member
How to you measure the velocity of an electron, You pass a photon through it and measure the frequency change to know the probability density. Well, that action alone induces energy and increase in energy leads to increase in momentum
Heisenberg probably used a similar argument to justify the original uncertainty principle, but the modern uncertainty relation is a theorem in QM, and its proof looks nothing at all like this argument. See this post.

The basic idea is still there..the only problem is that heinsenberg did not really understand what was going on so the argument of experiment became a plausible explanation. ..http://en.wikipedia.org/wiki/Uncertainty_principle here is the proof.
What QM only says is that instead of considering a particle as a particle.. lets take it as a wave function. And since the wavefunction does not have an amplitude, the position becomes a probability. It is all in the language there is no contradiction. And by considering it as a wave function, then the argument of the uncertainty depending on the instrument is quashed.

The experiment is not valid even classically. You cannot make two position measurements and conclude that the velocity is *exactly* $$\Delta x / \Delta t$$.

Fredrik
Staff Emeritus
Gold Member
The experiment is not valid even classically. You cannot make two position measurements and conclude that the velocity is *exactly* $$\Delta x / \Delta t$$.
Yes, but in QM this isn't even approximately correct (if the position measurements are accurate enough), and the situation gets worse, not better, when we measure the positions more accurately.

Thanks everyone for helping me out! :D I'm only a beginner in QM and only starting to grasp the principles. I'm not sure if I understood it completely, but I think I'm on the way.:)

I have another doubt regarding the measurement of the position of an electron. Why do we need light to ''see'' it? Can't we use some sort of a (although not feasible perhaps) motion detector, or some other instrument that doesn't excite the electron?

a motion detector makes use of a specific frequncy of light...so 'Light' is still a factor. What we use light is because that is the only way we know. So the scattering of light by an electron, will determine the possible position of the particle, In short.

I'm still not convinced that light is a necessary factor. for instance,when an electron moves, it creates some disturbance around it. Isn't it possible to detect this disturbance by using an instrument, like in the real world analogy, a wind-speed measuring instrument or something of that sort that doesn't utilise light for measurement?

I'm still not convinced that light is a necessary factor. for instance,when an electron moves, it creates some disturbance around it.

Yes, the electron can emit a photon. Apart from that, can't there be other forms of disturbances?

Fredrik
Staff Emeritus
Gold Member
In principle, you could use gravitational interactions, although it would be impossible to carry out in practice. But the uncertainty relations have nothing to do with what sort of "disturbances" you use. Just look at the proof.

I had a thought going in my head today about the uncertainty principle. Forgive me if it sounds too silly.

Consider an electron in motion. Now, suppose that I'm measuring it's position with infinite accuracy. So, by the uncertainty principle, it's velocity is blurry. But what if I measure the position of the electron again, after a very short interval of time, with infinite accuracy? Then, I can compare the distance it has moved from its previous position, and as I know the time taken for the electron to move this distance, i can figure out its velocity. So, I can measure both the position and velocity of the electron simultaneously, thereby violating the uncertainty principle.

So, what is the mistake I've made in this theoretical experiment?

or measure first its velocity and with circular trap sligthy after, its position, then you have both.

Drakkith
Staff Emeritus
or measure first its velocity and with circular trap sligthy after, its position, then you have both.

Nope. You have made 2 different measurements. In each one you are only measuring one thing accurately, not the other.

dextercioby
Homework Helper
My take on this ever-occuring issue in this QM subforum is this one spelled below:

The so-called <uncertainty principle> for me is nothing by a theorem in applied functional analysis which needs to be <blended> with one of the long-disputed interpretations of some properly chosen set of mathematical statements. Or with all, if possible.

In the post #1 of this thread https://www.physicsforums.com/showthread.php?t=470380 I have offered for public debate a set of axioms which would provide a proof for the theorem, if the rigor of functional analysis would be constantly kept.

So what's the physics behind this theorem applied for position and momentum ? My take on this is:

Assume one has been able to prepare an infinit set of identical systems in the same PURE physical state described by a point in the projective HS of all possible pure states. Futhermore, assume one wishes to measure on all this infinite set the observables position and momentum. One will obtain an infinite discrete set of values, call them x_1,..., x_inf, p_1,...,p_inf. This mathematical theorem asserts that the product of the 2 standard deviations is never smaller than h_bar/2. In other words, not all measured values for position and momentum are identical among themselves, respectively.

From my perspective, it's not a statement of whether we can measure position or momentum accurately, at the same time or not. We can measure these 2 observables with the greatest or even infinite technological precision on one or on several systems or on all systems from the ensemble, but for the measured values, the statistics pops in and asserts that neither of the 2 deviations can be 0. As a conclusion, even if the individual measurements were done with inifinite precision (thus accurate), we would still get a non-zero statistical dispersion from the mean values. Which means that the so-called HUP is not a statement about confirming our humanly finite precision measurement abilities, but rather of the special place these 2 observables bear in quantum mechanics. Which quantum mechanics? Precisely the one which leads us to the theorem by the formulated axioms and offers this interpretation of the mathematical theorem in question.

Question: Can I measure the position of an electron and its momentum at the same time or one after the other accurately ? Answer: Yes, I can. But I cannot be sure if the values I'm getting on the 2 screens for these 2 observables are the ones the systems really has, even if I used an infinite precision measurement.

Please, offer valid counterarguments to my statements, if any.

Last edited:
Nope. You have made 2 different measurements. In each one you are only measuring one thing accurately, not the other.

the relevant issue is if the electron have the 2 values simultaneously, not that it can't be measured at the same time, consequently refuting Counterfactual Definiteness (or prove CFD).
that way Counterfactual Indefiniteness is just epistemic.

Last edited:
Consider an electron in motion. Now, suppose that I'm measuring it's position with infinite accuracy. But what if I measure the position of the electron again, after a very short interval of time

or measure first its velocity and with circular trap sligthy after, its position, then you have both.

that is known as Two Vector formulation of Quantum Mechanics.

http://www.tau.ac.il/~yakir/yahp/yh30

...A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first preselected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time......

and

http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.3139v3.pdf

...A quantum state is not discernible by means of a single replica, but can be reconstructed only by performing
many measurements on identically prepared systems......

Last edited:
I had a thought going in my head today about the uncertainty principle. Forgive me if it sounds too silly.

Consider an electron in motion. Now, suppose that I'm measuring it's position with infinite accuracy. So, by the uncertainty principle, it's velocity is blurry. But what if I measure the position of the electron again, after a very short interval of time, with infinite accuracy? Then, I can compare the distance it has moved from its previous position, and as I know the time taken for the electron to move this distance, i can figure out its velocity. So, I can measure both the position and velocity of the electron simultaneously, thereby violating the uncertainty principle.

So, what is the mistake I've made in this theoretical experiment?

The Heisenberg Uncertainty principle can be expanded into the Feynman's Sum Over Histories, where an object, especially small particles, do not have only one history but every possible history. Your argument can be explained using this. In your first measurement, in every history, the position of the electron remains the same, whereas the electron's velocity in various histories is every single number! There is no such number which is not the velocity of an electron in your first measurement. In your second measurement, in all the histories of your electron, the same happens. But you don't know what history to use to find this velocity! It is every single velocity!

If in the 2nd measurement, you instead measure the velocity of the electron with infinite accuracy, the histories sort of twist. Instead of it having every single number as its velocity, it will have every single number as its position and only one as its velocity. In this case, you are in fact measuring different histories. Even in this case, you cannot compare.

I had a thought going in my head today about the uncertainty principle. Forgive me if it sounds too silly.

Consider an electron in motion. Now, suppose that I'm measuring it's position with infinite accuracy. So, by the uncertainty principle, it's velocity is blurry. But what if I measure the position of the electron again, after a very short interval of time, with infinite accuracy? Then, I can compare the distance it has moved from its previous position, and as I know the time taken for the electron to move this distance, i can figure out its velocity. So, I can measure both the position and velocity of the electron simultaneously, thereby violating the uncertainty principle.

So, what is the mistake I've made in this theoretical experiment?

The Heisenberg Uncertainty principle can be expanded into the Feynman's Sum Over Histories, where an object, especially small particles, do not have only one history but every possible history. Your argument can be explained using this. In your first measurement, in every history, the position of the electron remains the same, whereas the electron's velocity in various histories is every single number! There is no such number which is not the velocity of an electron in your first measurement. In your second measurement, in all the histories of your electron, the same happens. But you don't know what history to use to find this velocity! It is every single velocity!

If in the 2nd measurement, you instead measure the velocity of the electron with infinite accuracy, the histories sort of twist. Instead of it having every single number as its velocity, it will have every single number as its position and only one as its velocity. In this case, you are in fact measuring different histories. Even in this case, you cannot compare.