Bound states of Dirac electron in linear and parabolic potentials

When the mass term M (scalar-like potential) and the electrostatic potential V (vector-like potential) in the radial Dirac equation are linear (parabolic), the accurate and full spectra obtained by the sectioned series expansion method are reported. The evolution of the energy levels with M and V is observed, along with the competing effects of M and V on the bound states. The introduction of a magnetic field (similar to the tensor potential) enriches the variation of the energy levels. When M = V for the parabolic case, analytical results are demonstrated, in mutual agreement with the numerical ones. This work provides a comprehensive understanding of the relativistic quantum confinement of Dirac electrons, with implications for engineering electronic properties in Dirac materials through external fields.