Abstract
We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.
Highlights
Description and interpretation of coupled systems are of particular interest in physics because the interaction caused by coupling is responsible for novel quantum effects such as entanglement [1,2] and quadrature squeezing [3]
Coupled oscillatory systems can be used as a model to describe the interactions between atoms in a one-dimensional crystal with spring-like forces under white noise excitations [9,10]
We treated the application of the dynamical invariant on quantization of the system
Summary
Description and interpretation of coupled systems are of particular interest in physics because the interaction caused by coupling is responsible for novel quantum effects such as entanglement [1,2] and quadrature squeezing [3]. You can see an example of nano-optomechanical three coupled oscillators from Figure 1 In this contribution, we will study quantum dynamical invariant for dissipative three coupled oscillators based on exact quantum description of the system. We will study quantum dynamical invariant for dissipative three coupled oscillators based on exact quantum description of the system Such invariant can be used in analyzing various mechanical properties of the coupled oscillatory systems [19,20]. Our paper is structured as follows: In Section 2, we will represent the method for treating three coupled oscillators from preliminary level of mechanics. The classical and quantum invariant quantities for the three coupled oscillators will be derived based on the fundamental Hamiltonian dynamics.
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