B Uncertainty Principle & Non-Commuting Observables

[Anupama]

Do all observables which do not commute generate an uncertainty principle ?

 

[lekh2003]

The uncertainty principle applies to all observables. The act of knowing that the object does not commute makes you unsure about where it is.

 

[Fra]

Do all observables which do not commute generate an uncertainty principle ?

Yes you can put it like that, depending on what mean by uncertainty principle.

That observables does not commute simply means that they aren't independent. The details of this dependence can vary depending on what observables we talk about. "Conjugate variables" are related by means of the Fourier transform and is what one most commonly refers to. But in principle one can imagine any relation between independent variables that will imply some kind of "generalized uncertainty relation" that described a relation that constraints their mutual possible values, but not necessarily a simple one like for the case of x and p.

/Fredrik

 

[PeroK]

Do all observables which do not commute generate an uncertainty principle ?

Yes, the general uncertainty principle relates the uncertainty to the commutator of the two observables:

$\sigma_A^2 \sigma_B^2 \ge (\frac{1}{2i}\langle [\hat{A}, \hat{B}] \rangle)^2$