The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergence

For the traditional physical model of the quantum quartic anharmonic oscillator with the Hamiltonian H = ½(p2 + x2) + λx4, which plays a significant role in quantum field theory, elementary particle physics, and nuclear physics, its physical characteristics and properties are comprehensively studied and calculated. The method we propose for studying the model, based on expanding the system's wave function in a complete set of harmonic oscillator eigenfunctions, facilitates a thorough analysis and evaluation of all parameters and features of the corresponding quantum systems. This model is also widely used for studying molecular vibrations, phonon modes in solids, nonlinear optical phenomena, and more. We have calculated and constructed the wave functions of the anharmonic oscillator for various values of the oscillator coupling constant λ. Furthermore, an improved and modified expansion method, using a generalized optimizing oscillator basis with variable frequency, has also been proposed and studied in detail. This improved method drastically accelerates the convergence of expansions across the entire range of the coupling constant variation, thereby substantially increasing the efficiency of the applied method by allowing calculations with a very small number of expansion basis functions N ≲ 10. Consequently, this modified approach provides a practically complete, quite simple, and efficient solution to the problem of the quartic anharmonic oscillator, enabling the 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 the coupling constant.