Planck length challenges non-relativistic quantum mechanics of large masses

With the simplest proof ever, we justify the significance of quantum-gravity in non-relativistic quantum mechanics together with the related theories and experiments. Since the de Broglie wave length is inverse proportional to the mass, it would descend towards and below the Planck scale 10⁻³³ cm for large masses even at slow non-relativistic motion. The tricky relationship between gravity and quantum mechanics —well-known in the relativistic case—shows up in non-relativistic motion of massive objects. Hence the gravity-related modification of their Schrüdinger equation is mandatory. We also recall the option of an autonomous Newtonian quantum-gravity, a theory parametrized by ℏ and G. On cancellation of c from the Newtonian limit of Planck scale metric fluctuations is given a new hint. (paper)