  Elements are given of a calculation that identifies the size of a proton in
the Schroedinger equation for lepton-proton bound states, using the
renormalization group procedure for effective particles (RGPEP) in quantum
field theory, executed only up to the second order of expansion in powers of
the coupling constant. Already in this crude approximation, the extraction of
size of a proton from bound-state observables is found to depend on the lepton
mass, so that the smaller the lepton mass the larger the proton size extracted
from the same observable bound-state energy splitting. In comparison of
Hydrogen and muon-proton bound-state dynamics, the crude calculation suggests
that the difference between extracted proton sizes in these two cases can be a
few percent. Such values would match the order of magnitude of currently
discussed proton-size differences in leptonic atoms. Calculations using the
RGPEP of higher order than second are required for a precise interpretation of
the energy splittings in terms of the proton size in the Schroedinger equation.
Such calculations should resolve the conceptual discrepancy between two
conditions: that the renormalization group scale required for high accuracy
calculations based on the Schroedinger equation is much smaller than the proton
mass (on the order of a root of the product of reduced and average masses of
constituents) and that the energy splittings due to the physical proton size
can be interpreted ignoring corrections due to the effective nature of
constituents in the Schr\"odinger equation.
