The caldirola-kanai model and its equivalent theories for a damped oscillator

Based on the Caldirola-Kanai Hamiltonian, we first construct its equivalent theories of interest. In an equivalent theory, the Hamiltonian and the relation between the two Hubert spaces are obtained by performing a quantum canonical transformation. Then, we use the path-integral technique directly to calculate the exact propagators of the theories. The properties of the Caldirola-Kanai model, including the time evolution of the given initial wave functions and the expectation values of the physical quantities of interest, are studied by using the obtained propagators.