Response to: [Comment on Quantization of the damped harmonic oscillator [Serhan et al, J. Math. Phys. 59, 082105 (2018)]]

Abstract

This is a response to a recently reported comment [1] on paper [J. Math. Phys.59, 082105 (2018)] regarding the quantization of damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of [2] is incorrect, and thus the subsequent steps via the Nikiforov-Uvarov method are affected, and the energy eigenvalues should have been complex. However, we show here that the Hermiticity of the Hamiltonian should be firstly achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can be reached using the symmetrization rule. Applying the canonical quantization on the resulted Hermitian Hamiltonian and then using the Nikiforov-Uvarov method correctly, the energy eigenvalues will be real and exactly as given by Eq. (35) of [2].