  We construct an effective commutative Schr\"odinger equation in Moyal
space-time in $(1+1)$-dimension where both $t$ and $x$ are operator-valued and
satisfy $\left[ \hat{t}, \hat{x} \right] = i \theta$. Beginning with a
time-reparametrised form of an action we identify the actions of various
space-time coordinates and their conjugate momenta on quantum states,
represented by Hilbert-Schmidt operators. Since time is also regarded as a
configuration space variable, we show how an `induced' inner product can be
extracted, so that an appropriate quantum mechanical interpretation is
obtained. We then discuss several other applications of the formalism developed
so far.
