The classical and quantum mechanical correspondence for constant mass settings is used, along with some point canonical transformation, to find the position-dependent mass (PDM) classical and quantum Hamiltonians. The comparison between the resulting quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction and identification of the PDM-momentum operator. The same recipe is followed to identify the form of the minimal coupling of electromagnetic interactions for the classical and quantum PDM-Hamiltonians.

Using azimuthally symmetrized cylindrical coordinates, we consider some positiondependent mass (PDM) charged particles moving in position-dependent (PD) magnetic and Aharonov-Bohm flux fields. We focus our attention on PDM-charged particles with m ( − → r ) = g (ρ) = η f (ρ) exp (−δρ) (i.e., the PDM is only radially dependent) moving in an inverse power-law-type radial PD-magnetic fieldsUnder such settings, we consider two almost-quasi-free PDM-charged particles (i.e., no interaction potential, V ( − → r ) = 0) endowed with g (ρ) = η/ρ and g (ρ) = η/ρ 2 . Both yield exactly solvable Schrödinger equations of Coulombic nature but with different spectroscopic structures. Moreover, we consider a Yukawa-type PDM-charged particle with g (ρ) = η exp (−δρ) /ρ moving not only in the vicinity of the PD-magnetic and Aharonov-Bohm flux fields but also in the vicinity of a Yukawa plus a Kratzer type potential force field V (ρ) = −V• exp (−δρ) /ρ−V 1 /ρ+V 2 /ρ 2 . For this particular case, we use the Nikiforov-Uvarov (NU) method to come out with exact analytical eigenvalues and eigenfunctions. Which, in turn, recover those of the almost-quasi-free PDM-charged particle with g (ρ) = η/ρ for V• = V 1 = V 2 = 0 = δ.

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Analogous to Landau quantization related to a neutral particle possessing an electric quadrupole moment, we generalize such a Landau quantization to include position-dependent mass (PDM) neutral particles. Using cylindrical coordinates, the exact solvability of PDM neutral particles with an electric quadrupole moment moving in electromagnetic fields is reported. The interaction between the electric quadrupole moment of a PDM neutral particle and a magnetic field in the absence of an electric field is analyzed for two different radial cylindrical PDM settings. Next, two particular cases of radial electric fields ( − → E = λ ρ ρ and − → E = λρ 2 ρ) are considered to investigate their influence on the Landau quantization (of this system using the same models of PDM settings). The exact eigenvalues and eigenfunctions for each case are analytically obtained.