Quantization of the Bateman damping system with conformable derivative

In this work, the conformable Bateman Lagrangian for the damped harmonic oscillator system is proposed using the conformable derivative concept. In other words, the integer derivatives are replaced by conformable derivatives of order [Formula: see text] with [Formula: see text]. The corresponding conformable Euler–Lagrange equations of motion and fractional Hamiltonian are then obtained. The system is then canonically quantized and the conformable Schrödinger equation is constructed. The fractional-order dependence of the energy eigenvalues [Formula: see text] and eigenfunctions [Formula: see text] is obtained using suitable transformations and the extended fractional Nikiforov–Uvarov method. The corresponding conformable continuity equation is also derived and the probability density and probability current are thus suitably defined. The probability density evolution as well as its dependence on [Formula: see text] is plotted and analyzed for various situations. It is found that the energy eigenvalues are real and there are sort of gradual ordering in the behavior of the probability densities.