  We note that if we construct from the observed anomalous acceleration
$a_P=(8.74\pm1.33)\times 10^{-10}m/s^2$ of the spacecrafts Pioneer 10 and 11
towards the Sun, the proton mass $m_p$ and Newton's gravitational constant $G$
a quantity $\ell_P:=(Gm_p/a_P)^{1/2}$, which has the same dimension as length,
then $\ell_P$ is roughly of the same order of magnitude as is the Compton wave
length of a proton. We formulate a simple quantum mechanical hypothesis, which
aims to provide an explanation to this interesting numerical coincidence.
