  Noncommutivity of position and momentum makes it difficult to formulate the
unambiguous structure of the kinetic part of Hamiltonian for the
position-dependent effective mass (PDEM). Various existing proposals of writing
the viable kinetic part of the Hamiltonian for PDEM, conceptually lack from
first principle calculation. Starting from the first principle calculation, in
this article, we have advocated the proper self-adjoint form of the kinetic
part of Hamiltonian for PDEM. We have proposed that ambiguity of construction
of viable kinetic part for PDEM can be avoided if one takes the care from the
Classical level combination of position and momentum. \\ In the quantum level,
the spatial points do not appear in equivalent footing for the measure of
inertia (mass). This exhibits the existence of an inertia potential. Thus the
new structure of the Kinetic part differs from the existing structure of the
kinetic part of Hamiltonian by providing an extra potential like contribution.
This inertia potential can be absorbed with the external potential and redefine
the known structure of PDEM under this effective potential. This enables us to
apply the existing formalism of quantum mechanics. The coherent state
structures for the newly proposed form of Hamiltonian are provided for a few
simple experimentally important models.
