The quantum damped harmonic oscillator

Starting with the quantization of the Caldirola–Kanai Hamiltonian, various phenomenological methods to treat the damped harmonic oscillator as a dissipative system are reviewed in detail. We show that the path integral method yields the exact quantum theory of the Caldirola–Kanai Hamiltonian without violation of Heisenbergʹs uncertainty principle. Through the dynamical invariant and second quantization methods together with the path integral, we also present systematically the exact quantum theories for the various dissipative harmonic oscillators, bound and unbound quadratic Hamiltonian systems, and the relation between the canonical and unitary transformations for the classical and quantum dissipative systems.