Heisenberg uncertainty question

[lightoflife]

I read this article by Hawking which includes this quote "
In classical mechanics one can
predict the results of measuring both the
position and the velocity of a particle.
In quantum mechanics the uncertainty
principle says that only one of these
measurements can be predicted; the ob
server can predict the result of measur
ing either the position or the velocity but
not both. Alternatively he can predict
the result of measuring one combination
of position and velocity."

Can someone elaborate on the last sentence "Alternatively he can predict
the result of measuring one combination of position and velocity"

How is that supposed to work exactly? If you can know only one or the other then how do you measure the combination to predict anything?

 

[bhobba]

I don't know what Hawking meant. I can however tell you what the correct statement of the principle is.

Suppose you have a large number of similarly prepared systems ie all are in the same quantum state. Divide them into two equal lots. In the first lot measure position to a high degree of accuracy. QM places no limit on that accuracy - its a misunderstanding of the uncertainty principle thinking it does. The result you get will have a statistical spread. In the second lot measure momentum to a high degree of accuracy - again QM places no limit on that. It will also have a statistical spread. The variances of those spreads will be as per the Heisenberg Uncertainty principle.

Thanks
Bill

 

[DrChinese]

I read this article by Hawking which includes this quote "
In classical mechanics one can
predict the results of measuring both the
position and the velocity of a particle.
In quantum mechanics the uncertainty
principle says that only one of these
measurements can be predicted; the ob
server can predict the result of measur
ing either the position or the velocity but
not both. Alternatively he can predict
the result of measuring one combination
of position and velocity."

Can someone elaborate on the last sentence "Alternatively he can predict
the result of measuring one combination of position and velocity"

How is that supposed to work exactly? If you can know only one or the other then how do you measure the combination to predict anything?

Yes, this scenario is actually an application of the HUP. It is the product of 2 non-commuting observables' standard deviations which cannot be less than a certain quantity (a constant). A measurement (say p) which is not intended to be more accurate than a certain value can be executed. Another (say q) which is also not intended to be more accurate than a certain value can then be executed. As long as those are properly executed, you will know a combination of p and q as Hawking says.

This could, for example, be done on entangled particles: execute a measurement of p on Alice, execute a measurement of q on Bob. Done properly, you would not violate the HUP and you would still know a lot about Alice (and Bob). You would know p +/- and you would know q +/-.

Again, you could not execute such measurements with more precision than the HUP allows and expect useful information. This is *not* the EPR example, because in that example non-commuting observables are measured to very tight precision. So the proper execution in my example means the margin of error was intentionally made larger.

 

[soothsayer]

This is a disappointingly misleading quote from Hawking. There is nothing to stop you from measuring both the position and momentum of a particle, as Hawking seems to imply at first. The HUP states that you cannot simultaneously know both of them at an arbitrary level of accuracy.