The quantum propagator of the harmonic oscillator with damping A white noise path integral approach - Cubero, Ryan John A.; Bornales, Jinky B. (2011)
Recently, the infinite dimensional white noise calculus by T. Hida and L. Streit has successfully provided a more mathematically well-defined formulation to the Feynmann path integral. It has already been successful in solving quantum problems such as the perturbed harmonic oscillator and systems with exponentially-growing potentials. In this paper, white noise analysis is utilized to solve the quantum propagator, expressed as the Feymann path integral, for the harmonic oscillator with damping described by the Caldirola-Kanai Lagrangian, L=1/2 mexp(γt) ẋ^2-1/2 mexp(γt) v^2 x^2 for a particle of mass m and angular frequency v, which is subjected to a damping of constant γ. In this approach, a parametrization of the paths of the particles in terms of an exponentially decreasing Brownian motion was employed thereby formulating the Feymann path integral in the context of white noise analysis where the S- and T- transforms in white noise calculus are then used to evaluate the integral. The calculated propagator agrees with known results. The time-dependent wave function for the harmonic oscillator with damping is then extracted via the symmetrization of the calculated propagator which also agrees with known results.
Keywords: Noise; Mathematical models