Does the Uncertainty Principle Make Precise Momentum Measurement Impossible?

[luxiaolei]

Hi all, I am sure this is a quite simple question but I just can't figure it out, any helps would be greatly appreciated.

If a particle stay in a state in a very short time, then by energy time uncertainty relation, it's energy must has a great uncertainty. Also because of energy is propotion to momentum, then does it mean the momentum is uncertained aswell?So one measure it's momentum percisely just became impossible?

 

[DiracRules]

I can't answer with 100% certainty, but I can tell you something that could clarify something.
when we did this, our professor said that the energy-time indetermination principle has to be interpreted in a different way than the position-momentum one: while the latter refers to dynamic variable, so it can be interpreted as "we can't measure both position and momentum with infinite precision", the former takes into account time, which, at least in non-relativistic quantum mechanic, is not a dynamic variable, or at least is treated differently from momentum and space - just think that in Schroedinger's equation (non relativistic) time appears in a first partial derivative while space in a second partial derivative, so they are not treated equally.
So, the energy-time indetermination principle has to be interpreted as "a state that has a certain indetermination on the energy can exist for a certain amount of time"; some examples of application can be the following:
a state with fixed energy ($\Delta E=0$) is a stationary state ($\Delta t=\infty$);
a particle that has a greater mass than another lives less.

I know that I haven't answer directly to your question, but I actually don't know the correct answer, so I was just putting down few ideas :D :D

 

[Naty1]

Not a simple question!

In essence, if you want an accurate/precise series of measurements of momentum (at various points in time) you can get them to arbitrary precision. You don't have that ability with time/energy and momentum/position pairs.

In simple terms, Heisenberg uncertainty applies to certain characteristics whose values do not commute: energy and time are one pair; position and momentum another. One way to view the implication of this is that if you measure one before the other, then reverse the order of measurement you'll get a different measurement. But that is not the whole story.


Here is an essential component of a very long discussion on (almost) your question (I think that discussion is still being argued):


This is from Zapper:

The HUP isn't about a single measurement and what can be obtained out of that single measurement. It is about how well we can predict subsequent measurements given the identical conditions.

and

What I am trying to get across is that the HUP isn't about the knowledge of the conjugate observables of a single particle in a single measurement. I have shown that there's nothing to prevent anyone from knowing both the position and momentum of a particle in a single mesurement with arbitrary accuracy that is limited only by our technology. However, physics involves the ability to make a dynamical model that allows us to predict when and where things are going to occur in the future. While classical mechanics does not prohibit us from making as accurate of a prediction as we want, QM does!

http://physicsandphysicists.blogspot.com/2006/11/misconception-of-heisenberg-uncertainty.html


Somebody in the recent past posted this...my boldface.. (I did not record the poster, maybe even Zapper??..was a trusted source here.) I'm posting this to confirm that it is an equivalent description, that it matches Zappers blog...

...

to measure a particle's momentum, we need to interact it with a detector, which localizes the particle. So we actually do a position measurement (to arbitrary precision). Then we calculate the momentum, which requires that we know something else about the position of the particle at an earlier time (perhaps we passed it through a narrow slit). Both of those position measurements, and the measurement of the time interval, can be done to arbitrary precision, so we can calculate the momentum to arbitrary precision. From this you can see that in principle, there is no limitation on how precisely we can measure the momentum and position of a single particle.

Where the HUP comes into play is that if you then repeat the same sequence of arbitrarily precise measurements on a large numbers of identically prepared particles (i.e. particles with the same wave function, or equivalently particles sampled from the same probability distribution), you will find that your momentum measurements are not all identical, but rather form a probability distribution of possible values for the momentum. The width of this measured momentum distribution for many particles is what is limited by the HUP. In other words, the HUP says that the product of the widths of your measured momentum probability distribution, and the position probability distribution associated with your initial wave function, can be no smaller than Planck's constant divided by 4 times pi

Here are two more complementary explanations from that discussion:


atyy:

The uncertainty relation is defined as the non-commutation of position and momentum operators. A state with definite momentum is an eigenstate of momentum, and state with definite position is an eigenstate of position. The commutation relation prevents an eigenstate of momentum from being an eigenstate of position, so there is no state with definite momentum and position, and so it cannot be prepared

.

Originally Posted by fuesiker:

The uncertainty principle has nothing to do with observation. It does not arise due to observation. It is the nature of things that there are non-commuting observables (time and energy, momentum and position) that cannot be determined simultaneously to arbitrary precision.

 

[Naty1]

ok, here is the very long discussion from which I made the excerpts in my previous post:

what is it about position and momentum that forbids knowing both quantities at once?

https://www.physicsforums.com/showthread.php?t=516224


Reading the first dozen or two dozen posts there might be worthwhile...

 

[luxiaolei]

@DiracRules

Thank you for your replay, and it's indeed a great help. Now I am just waiting for more people to come and bring more ideas:) thanks again.