Study of the quartic anharmonic oscillator using the nonlinear Rayleigh - Ritz variational method. I. Calculation and analysis of energy levels and wave functions
Abstract
For the quantum quartic anharmonic oscillator with Hamiltonian H = ½(p2 + ω2x2) + λx4, our study has been further developed using a convergent expansion of the system's wave function in a modified oscillator basis with an adjustable frequency ω0, i.e., in the complete set of eigenfunctions {φn(ω0;x)} of a reference harmonic Hamiltonian H(0)(ω0) = ½(p2+ω02x2). Our proposed approach enables a significant improvement in the convergence rate of expansions through the additional adaptation of the basis functions to the specific structure of the system being analyzed. Primary attention is given to the rigorous mathematical implementation of the variational scheme, treating the basis frequency ω0 as a true nonlinear variational parameter subject to optimization. Within the framework of the generalized nonlinear Rayleigh - Ritz variational method, the system's energy is minimized with respect to the nonlinear parameter ω0 to accurately determine the optimal value of ω0 for different values of the oscillator coupling constant λ and the variational basis size N. This study provides, for the first time, a fully rigorous numerical implementation of this procedure for the quartic anharmonic oscillator, ensuring precise control and enabling a transition from heuristic parameter selection to a systematic and reproducible algorithmic approach. This rigorous variational justification enables the exploration of the energy functional's structure and the behavior of the nonlinear parameter. It…
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