  A classical result due to Paley and Wiener characterizes the existence of a
non-zero function in $L^2(\mathbb{R})$, supported on a half line, in terms of
the decay of its Fourier transform. In this paper we prove an analogue of this
result for compactly supported continuous functions on the Euclidean motion
group $M(n)$. We also relate this result to a uniqueness property of solutions
to the initial value problem for time-dependent Schr\"odinger equation on
$M(n)$.
