Math of the uncertainty principle

[Durandarte]

How can one operate the deltas of the uncertainty relation? I know they're supposed to be the standard dev, but how are they operated in physical reality? Is there some sort of function to make them have a physical meaning?

 

[sanjib ghosh]

As you know (<ψ|ΔA|ψ>)²=<ψ|AA|ψ>-(<ψ|A|ψ>)².
This is the way ΔA act on some state |ψ>.

 

[jarod765]

Physically the $\Delta$ represents the variance of a distribution (the width of the probability distribution). The variance gives a measure of how likely one is to deviate from the average value.

 

[ImaLooser]

How can one operate the deltas of the uncertainty relation? I know they're supposed to be the standard dev, but how are they operated in physical reality? Is there some sort of function to make them have a physical meaning?

I never much cared for this simple relation. I think it is much more interesting to know that the momentum probability distribution is the Fourier transform of the position probability distribution. If one distribution is spiky then the other is very broad. This is both more general and more informative, and avoids that word "certainty" which IMO mystifies the situation.

In other words, there is a strict one-to-one mapping between these two distributions. Each completely defines the other. If I were to pick one fact of quantum mechanics deserving of wider recognition, this would be it.