The effective Hamiltonian and propagator of a parabolic confined dissipative electron under a perpendicular magnetic field

We study the model of an electron in an isotropic quantum dot in the presence of a perpendicular magnetic field and dissipation. Starting from the system plus bath approach, the quantum Langevin equations are first obtained and consequently the equation of motion. Then the effective Hamiltonian is derived from the equation of motion. We obtained a new effective Hamiltonian for the case when the magnetic field and dissipation are both present. This effective Hamiltonian is quite different from the one used widely for time-dependent harmonic oscillators under a magnetic field in earlier studies. The corresponding propagator for this new effective Hamiltonian is also calculated. As an example of application, the dissipative dynamics of a Gaussian wavepacket confined in small quantum dots in the presence of a perpendicular magnetic field is studied.