What is the relationship between uncertainties in position and momentum?

[ehanes7612]

Homework Statement

Homework Equations

The Attempt at a Solution

 

[ehanes7612]

I understand the uncertainty principle (delta x)(delta p) is approximately h (planck's constant) in definition, but I am having trouble seeing that if you measure the certainty to one aspect of a particle, that you won't be able to measure as accurately the certainty of another ..for example..
regarding single slit experiment..I understand how to derive the fringe width using delta x (slit width) and some basic trig.
I also understand how to derive momentum using the transverse momentum (delta p) of the particles as they hit the screen.
And when asked to use the uncertainty principle to show that h is approximate to (delta x)(delta p) ..its an easy algebraic step..
but I am not seeing how this represents an understanding that if you measure one with certainty, the other can't be measured with the same certainty
Is this implicit in Heisenberg's expression that only becomes clear in running the experiment ? I feel like I am missing something really basic here

BTW, this is not a homework problem ..I am on school break and trying to wrap my head around this before going on to quantum mechanics..I figure I need to get this basic concept down ASAP.

 

[CWatters]

I am not seeing how this represents an understanding that if you measure one with certainty, the other can't be measured with the same certainty

As I recall the delta in delta X is the uncertainty or standard deviation of X.

So the equation is essentially saying...

The uncertainty of X * The uncertainty of p => a lower limit

Therefore if you try and reduce the uncertainty of X then the uncertainty of p must get larger or you would break the inequality.

See also..

but don't miss the note at the end.

 

[CWatters]

PS. I forgot to add that..

Because it's a lower limit you can reduce the uncertainty of both measurements but only to a point. After that any further reduction in one makes the other larger.

 

[phinds]

I've seen statements on this forum supporting very strongly the point of view that you cannot measure both simultaneously to an arbitrary degree of precision and I've seen other statements, equally strong, saying yes you can.

What is NOT in dispute is that it is impossible to make repeated measurements of identically set up situations that get exactly the same response. That is, if you create situations in QM that classically you would absolutely expect to produce identical results, you will not see identical results, you will see a probabilistic distribution of results. That's basically exactly what the single slit experiment shows, and it also shows that the variance in results is a function of how precisely you make the measurements.

 

[ehanes7612]

Thanks, your explanations helped quite a bit..and thanks for the video

 

[haruspex]

I'd like to clarify an aspect of your question...

I understand how to derive the fringe width using delta x (slit width)

I also understand how to derive momentum using the transverse momentum (delta p)

how this represents an understanding that if you measure one with certainty, the other can't be measured with the same certainty

Putting those together, it sounds like you are discussing the uncertainty in the deltas, not in the underlying variables.

 

[ehanes7612]

I'd like to clarify an aspect of your question...
Putting those together, it sounds like you are discussing the uncertainty in the deltas, not in the underlying variables.

well yeah..as something that denotes a change or the possible variation in the variable...that is as far as my understanding goes..a friend of mine (math graduate student) expounds on the concept in great detail but I haven't reached that level yet. From the responses and the video (and my limited knowledge of analysis), my understanding is that the deltas depend greatly on the accuracy of your measurements, ...so although you could measure the delta x to a great deal of accuracy..you can't measure the delta p of one particle to the same accuracy given the range of delta p inherent in the experiment...that's my takeaway. But anything you want to add to make my understanding more sophisticated..I am all ears.

 

[haruspex]

well yeah..as something that denotes a change or the possible variation in the variable...that is as far as my understanding goes..a friend of mine (math graduate student) expounds on the concept in great detail but I haven't reached that level yet. From the responses and the video (and my limited knowledge of analysis), my understanding is that the deltas depend greatly on the accuracy of your measurements, ...so although you could measure the delta x to a great deal of accuracy..you can't measure the delta p of one particle to the same accuracy given the range of delta p inherent in the experiment...that's my takeaway. But anything you want to add to make my understanding more sophisticated..I am all ears.

The expression $\Delta \vec x. \Delta \vec p > h$ (or h-bar, or whatever) puts a limit on how accurately $\vec x$ and $\vec p$ can be known simultaneously. It does not put a limit on the accuracy of knowing $\Delta \vec x$, $\Delta \vec p$. The deltas are the uncertainties.