Pioneer Effect: An Interesting Numerical Coincidence

We note that if we construct from the observed anomalous acceleration $a_P=(8.74\pm 1.33)\times {10}^{-10}m\mathrm{/}{s}^2$ of the spacecrafts Pioneer 10 and 11 towards the Sun, the proton mass $m_p$ and Newton’s gravitational constant $G$ a quantity ${\mathrm{\ell }}_P:=(G{m}_p\mathrm{/}{a}_P{)}^{1\mathrm{/}2}$, which has the same dimension as length, then ${\mathrm{\ell }}_P$ 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.