Now to the concept which both fascinates and frustrates me. The Uncertainty Principle. I understand what its used for and all that, but i simply dont understand why it is true? You guys are my last hope… ive picked up quantam books and read chapters on the damn thing, but i still dont get it!
Um, by observing something you change it. Therefore you can’t know both its velocity and position at the same time.
I.E taking a snapshot photo of it tells you nothing of it’s speed. Only where it is. In tandem the photons used to take the picture have altered the particle in some way and is therefore traveling at a different velocity.
Basically the uncertainty principle (in it’s generalized form) states the minimum uncertainty between two observables, for what are known as complemantary pairs of observables this value is always non-zero and is greater than or equal to h-bar times a half, that is:
Postion and mometum:
ΔxΔp[sub]x[/sub] ≥ ħ/2
Energy and time:
ΔEΔt ≥ ħ/2
Orthogonal components of angular momentum:
Δω[sub]x[/sub]Δω[sub]z[/sub] ≥ ħ/2
etc.
For an observable ‘X’ the uncertainty in X, ΔX, represents the standard deviation in set of measuremnts of X. This means that there is a very definte limit to how well the behaviour of a system can be predicted.
Or more basically if I repeatedly perform an experiment to measure the postion and momentum of a particle, no matter how precise the experiment is I will always find that the standard deviation of the momentum mutplied by the standard deviation of the postion for these results will always be greater than or equal to h-bar time a half.
Quantum particles are considered to be particles. But between measurements the only knowledge we have of these particles is given by a wavefunction. A classical particle has a precise position whereas a classical wave is spread out and therefore does not. Maybe you can begin to see a possible problem here when trying to describe a particle via a wave
In order for an ensemble of quantum particles to have a precise position the wavefunction has to be sharply spiked in what is called a Dirac delta function. And in order to produce this condition you have to use an infinite range of frequencies. But the particle’s momentum is proportional to frequency and since the frequency range is infinite you can’t say anything at all about its momentum.
So if you know position precisely you can’t know anything about momentum and vice versa. A compromise wavefunction, which has a minimum uncertainty of both of these variables, is called a Gaussian and it looks a little like a parabola.