Response to “Comment on ‘Quantization of the damped harmonic oscillator’” [J. Math. Phys. 60, 094101 (2019)]

Abstract

This is a response to a recently reported comment [F. M. Fernández, J. Math. Phys. 60, 094101 (2019)] on paper [M. Serhan et al., J. Math. Phys. 59, 082105 (2018)] regarding the quantization of the damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)] 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 first achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can be reached using the symmetrization rule. The canonical quantization on the resulted Hermitian Hamiltonian yields the real energy eigenvalues as exactly given by Eq. (35) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)].