Position-dependent mass momentum operator and minimal coupling: point canonical transformation and isospectrality

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. It turned out that whilst the minimal coupling may very well inherit the usual form in classical mechanics (i.e., $p_j\left(\overrightarrow{x}\right)\to p_j\left(\overrightarrow{x}\right)-e{A}_j\left(\overrightarrow{x}\right)$, where $p_j\left(\overrightarrow{x}\right)$ is the j-th component of the classical PDM-canonical-momentum), it admits a necessarily different and vital form in quantum mechanics (i.e., ${\hat{p}}_j\left(\overrightarrow{x}\right)/\sqrt{m\left(\overrightarrow{x}\right)}\to \left({\hat{p}}_j\right(\overrightarrow{x})$$-e{A}_j\left(\overrightarrow{x}\right))/\sqrt{m\left(\overrightarrow{x}\right)}$, where ${\hat{p}}_j\left(\overrightarrow{x}\right)$ is the j-th component of the quantum PDM-momentum operator). Under our point transformation settings, only one of the two commonly used vector potentials (i.e., $\overrightarrow{A}\left(\overrightarrow{x}\right)\sim (-x_2,x_1,0)$) is found eligible and is considered for our Illustrative examples.