  In [J. Math. Phys. 51 (2010) 022104] a self-adjoint operator was introduced
that has the property that it indicates the direction of time within the
framework of standard quantum mechanics, in the sense that as a function of
time its expectation value decreases monotonically for any initial state. In
this paper we study some of this operator's properties. In particular, we
derive its spectrum and generalized eigenstates, and treat the example of the
free particle.
