Quantum theory of the harmonic oscillator in nonconservative systems

Starting with the quantization of the Caldirola-Kanai Hamiltonian, we review in detail various phenomenological methods to treat the damped harmonic oscillator as a dissipative system. We show that the path integral method yields the exact quantum theory of the Caldirola-Kanai Hamiltonian without violating the Heisenberg's uncertainty principle. Through the dynamical invariant together with the path integral. we also present systematically the exact quantum theories for various dissipative harmonic oscillators, as well as the relations between the canonical, and unitary transformations for the classical, and quantum dissipative systems