  The effect of the noise induced by gravitons in the case of a freely falling
particle from the viewpoint of an external observer has been recently
calculated in \href{https://link.aps.org/doi/10.1103/PhysRevD.107.066024}{Phys.
Rev. D 107 (2023) 066024}. There the authors have calculated the quantum
gravity modified Newton's law of free fall where the spacetime has been
considered to be weakly curved. In our work, we extend this work by calculating
the variance in the velocity and eventually the momentum of the freely falling
massive particle. From this simple calculation, we observe that the product of
the standard deviation in the position with that of the standard deviation in
momentum picks up a higher order correction which is proportional to the square
of the standard deviation in momentum. We also find out that in the Planck
limit (both Planck length and Planck mass), this uncertainty product gives the
well-known form of the generalized uncertainty principle. We then calculate a
similar uncertainty product when the graviton is in a squeezed state, and
eventually, we get back the same uncertainty product. Finally, we extend our
analysis for the gravitons being in a thermal state and obtain a
temperature-dependent uncertainty product. If one replaces this temperature
with the Planck temperature and the mass of the particle by the Planck mass,
the usual uncertainty product appears once again. We also obtain an upper bound
of the uncertainty product thereby giving a range of the product of the
variances in position and momentum.
