(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this study, we developed a Hamiltonian framework for the anisotropic harmonic oscillator and applied the Hamilton-Jacobi equations, which are particularly well-suited for this approach. We were able to find the action function by including boundary conditions in the study of a dissipative anisotropic harmonic oscillator. To quantize the system, we employed three distinct methods: the WKB approximation, canonical quantization, and the creation and annihilation operator technique. The quantization of the Heisenberg equations was achieved by reformulating them using Poisson bracket quantum variables. Through the application of creation and annihilation operators, we derived the energy levels and quantum states of the system. The results demonstrated complete consistency across the different methods. This model exhibits strong agreement among the solutions derived from various approaches, highlighting its importance in advancing the understanding of dissipative quantum systems.