Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A
and such vectors that the lower bound for the product of standard deviations
calculated for these vectors is zero:
. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices
and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in
–symmetric quantum theory and the problems associated with it are also studied.
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