"I consider the model of quantum damped oscillator with time-dependent frequency and damping coefficients, which can be used to predict the results of current experiments aimed at the discovering the so-called Dynamical Casimir Effect. It is based on the Heisenberg–Langevin approach to the description of open quantum systems. The main emphasis is made on the comparison of the exactly solvable special cases with general approximate solutions found earlier. The examples considered demonstrate that simple approximate formulas are quite reliable for the analysis of realistic experimental situations."
In this paper we extend the so-called dual or mirror image formalism and Caldirola's- Kanai's formalism for damped harmonic oscillator to the case that both frictional coefficient and time-dependent frequency depend on time explicitly. As an solvable example, we consider the case that frictional coefficient $ \ga (t) = \frac{ \ga_0}{1 + q t} , (q > 0 )$ and angular frequency function $ w(t) = \frac{ w_0}{ 1 + q t } $. For this choice, we construct the quantum harmonic Hamiltonian and express it in terms of
su(2) algebra generators. Using the exact invariant for the Hamiltonian and its unitary transform, we solve the time-dependent Schro\dinger equation with time-dependent frictional coefficient and time-dependent frequency.
In this research paper, we developed a theory and model of the forced simple harmonic oscillators in quantum mechanics system in a framework of a general approach to the Heisenberg picture. We can be used the time-dependent Hamiltonian for the forced simple harmonic oscillators. We use first-order ordinary linear differential equation to solve the time-dependent lowering operator and raising operator. We evaluate the lowering operator and raising operator as the function of time for particle mass depend on the parameter frequency of oscillation , the damping coefficient of particle bound in the simple harmonic oscillator potential, the initial force .