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We point out an erroneous handling in the literature regarding solutions of the ( 1 + 1 ) -dimensional Duffin–Kemmer–Petiau equation with linear potentials in the context of quantum mechanics with minimal length. Furthermore, using Brau's approach, we present a perturbative treatment of the effect of the minimal length on bound-state solutions when a Lorentz-scalar linear potential is applied.

We consider a soluble covariant extension of the two-dimensional Dirac oscillator (2D DO), which breaks the infinite degeneracy of the energy spectrum. The energy eigenvalues and the corresponding eigenstates of the system are obtained algebraically using chiral creation and annihilation operators. The effect of the coupling to an external constant transverse magnetic field is investigated. The connection with Jaynes-Cummings (JC) and Anti-Jaynes-Cummings (AJC) models of quantum optics, and other features of the system are also discussed.

We present a detailed study for the classical and the quantum motion of a relativistic massless particle in an inverse square potential. The classical treatment shows that it is possible to find trajectories for which the system has a positive total energy and yet the particle remains bound to the potential center. The quantum approach to the problem is based on the exact solution of the corresponding semirelativistic spinless Salpeter equation for bound states. The connection between the classical and the quantum descriptions is made via the comparison of the associated probability densities for momentum.

We solve the DKP equation with a pseudoscalar linear plus Coulomb-like potential in a two-dimensional space–time. The problem is mapped into an effective quadratic plus inversely quadratic potential in a Sturm–Liouville type equation. The exact energy spectrum and the corresponding normalized eigenstates are obtained. The particular case of the DKP oscillator is also discussed.

We present an exact analytical treatment of the semi-relativistic spinless Salpeter equation with a one-dimensional Coulomb interaction in the context of quantum mechanics with modified Heisenberg algebra implying the existence of a minimal length. The problem is tackled in the momentum space representation. The bound-state energy equation and the corresponding wave functions are exactly obtained.

We present an exact solution of the ( 1 + 1 ) -dimensional Dirac equation with vector and scalar linear potentials in the context of modified quantum mechanics characterized by the presence of a non-zero minimum uncertainty in position. The bound-states energy spectrum and the corresponding momentum space wavefunctions are exactly obtained. Our findings are compared with approximated results existing in the literature.