That Heisenberg Uncertainity Principle

[PeterPeter]

Does the Heisenberg Uncertainity Principle mean:
1) If a particle is confined within a length x then it must jiggle around with a momentum given by p ~ h/2x PI

OR

2) If we measure the position of a particle to an accuracy of x then its momentum will be uncertain by ~ h/2x PI

 

[bhobba]

It means neither.

It says if you have a large number of similarly prepared systems, measure the position on half of them, and momentum on the other half, then the variance of those results will be as per the uncertainly relations.

It is not a statement about uncertainly of measurement - you can measure momentum or position as accurately as you like - its a statistical statement about the outcomes of a large number of measurements.

Thanks
Bill

 

[moriheru]

The more general version of the uncertainty principle, derived by the cauchy schwarz inequality states thet the product of the expectation values of two observables is greater than or equivaleent to half the expectation value of the commutators of the observables. The Heisenberg uncertainty principle just uses the fact that the commutator of X the position and P the momentum is -ih(bar) and substitiues that in the equation. The equations you wrote are just reformations of the principle