Abstract
Driven and damped parametric quantum oscillator is solved by Wei-Norman Lie algebraic approach, which gives the exact form of the evolution operator. This allows us to obtain explicitly the probability densities, time-evolution of initially Glauber coherent states, expectation values and uncertainty relations. Then, as an exactly solvable model, we introduce the driven Cauchy-Euler type quantum parametric oscillator, which appears as self-adjoint quantization of the classical Cauchy-Euler differential equation. We discuss some typical behavior of this oscillator under the influence of external terms and give a concrete example.
Highlights
Quantum harmonic oscillator with explicitly time-dependent Hamiltonian is a fundamental model, which appears in many physical branches such as quantum optics, quantum fluid, ion-traps, and cosmology
Where μ(t) > 0, ω2(t), B(t) and D(t) are sufficiently smooth, real-valued parameters depending on time
We introduce the Cauchy-Euler type quantum parametric oscillator as another exactly solvable model, and briefly discuss some of its typical properties
Summary
Quantum harmonic oscillator with explicitly time-dependent Hamiltonian is a fundamental model, which appears in many physical branches such as quantum optics, quantum fluid, ion-traps, and cosmology. Where μ(t) > 0, ω2(t), B(t) and D(t) are sufficiently smooth, real-valued parameters depending on time. The Hamiltonian (2) can be written as time-dependent linear combination of X 1(t) − x1(t) ( x2(t) ) x1(t) q2 , where the functions x1(t) and x2(t) are two linearly independent homogeneous solutions of the classical equation of motion, x
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