  The standard uncertainty relations (UR) in quantum mechanics are typically
used for unbounded operators (like the canonical pair). This implies the need
for the control of the domain problems. On the other hand, the use of (possibly
bounded) functions of basic observables usually leads to more complex and less
readily interpretable relations. Also, UR may turn trivial for certain states
if the commutator of observables is not proportional to a positive operator. In
this letter we consider a generalization of standard UR resulting from the use
of two, instead of one, vector states. The possibility to link these states to
each other in various ways adds additional flexibility to UR, which may
compensate some of the above mentioned drawbacks. We discuss applications of
the general scheme, leading not only to technical improvements, but also to
interesting new insight.
