Study of the Quartic Anharmonic Oscillator Using the System's Wave Function Expansion in the Oscillator Basis

The quantum quartic anharmonic oscillator with the Hamiltonian $H=\frac{1}{2}\left( p^{2}+x^{2}\right) +\lambda x^{4}$ is a classical and fundamental model that plays a key role in various branches of physics, including quantum mechanics, quantum field theory, high-energy particle physics, and other areas. To study this model, we apply a method based on a convergent expansion of the system's wave function in a complete set of harmonic oscillator eigenfunctions -- namely, the basis of eigenfunctions $\varphi^{(0)}_n$ of the unperturbed Hamiltonian $H^{(0)}=\frac{1}{2}\left(p^{2}+x^{2}\right)$. 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 $\lambda$ values. We have computed the energies of the ground and the first six excited states for a broad range of the coupling constant $\lambda$, and also calculated and constructed the corresponding wave functions. Additionally, we propose and detail an improved version of the expansion method using a modified optimized oscillator basis with variable frequency. This modification significantly accelerates the convergence of expansions across the entire range of $\lambda$, thereby greatly enhancing the efficiency of the method and allowing accurate calculations with a very small number of expansion functions $N\lesssim 10$. As a result, this modified approach provides an essentially complete, simple, and efficient solution to the problem of the anharmonic oscillator, enabling straightforward computation of all its physical properties -- including the energies and wave functions of both ground and excited states -- for arbitrary values of $\lambda$.