Studies in the quantization of dissipative systems play a role in several areas of physics ranging from electrodynamics to chromodynamics. In spite of this, the problem of quantizing the simplest dissipative system, namely, the damped harmonic oscillator (DHO), has yet remained largely unsolved [1]. The primary reason for this may be attributed to an early observation made by Lanczos [2] who noted that the forces of frictions are outside the realm of variational principle although Newtonian mechanics has no difficulty to accommodate them. In the recent past, Riewe [3] formulated the Lagrangian and Hamiltonian mechanics of dissipative systems within the framework of fractional calculus. It may be an interesting curiosity to adapt the formalism of Riewe to deal with the problem of quantizing the DHO. However, we are interested to treat a simple variant of the DHO using usual traditional machineries of classical and quantum mechanics. The usual equation of motion for the one dimensional DHO is given by mx+ α x+ kx= 0,(1)

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