Study of the quartic anharmonic oscillator using expansion in the oscillator basis

Abstract

For the quantum quartic anharmonic oscillator with the Hamiltonian H = 1/2(p^2 + x^2) + λx^4, we apply a method based on the system’s wave function convergent expansion in a complete set of harmonic oscillator eigenfunctions, namely, the basis of eigenfunctions φ_n^(0) of the unperturbed Hamiltonian H^(0) = 1/2(p^2 + x^2). This approach enables a thorough analysis and calculation of the oscillator’s physical characteristics. We demonstrate very good convergence of all calculated quantities with respect to the number of basis functions included in the expansion, over a wide range of the parameter λ variation. We have computed the energies of the ground and the first six excited states for a broad range of the coupling constant λ. We also calculated and constructed the wave functions of the system for various λ values. Additionally, we propose and detail an improved and modified expansion method, using a modified optimizing oscillator basis with variable frequency. This method significantly accelerates the convergence of expansions across the entire range of λ variation, thereby substantially increasing the efficiency of the applied method by allowing calculations with a very small number of expansion functions N ≲ 10. This modified approach provides an essentially complete, simple and efficient solution to the problem of the anharmonic oscillator, enabling relatively easy computation of all its physical properties, including the energies of the ground and excited states, as well as the wave functions of these states, for any values of λ.