Λεπτομέρειες βιβλιογραφικής εγγραφής
| Τίτλος: |
ДОСЛІДЖЕННЯ БІКВАДРАТИЧНОГО АНГАРМОНІЧНОГО ОСЦИЛЯТОРА НА ОСНОВІ НЕЛІНІЙНОГО ВАРІАЦІЙНОГО МЕТОДУ РЕЛЕЯ–РІТЦА. I. РОЗРАХУНОК ТА АНАЛІЗ РІВНІВ ЕНЕРГІЇ ТА ХВИЛЬОВИХ ФУНКЦІЙ. |
| Alternate Title: |
STUDY OF THE QUARTIC ANHARMONIC OSCILLATOR USING THE NONLINEAR RAYLEIGH–RITZ VARIATIONAL METHOD. I. CALCULATION AND ANALYSIS OF ENERGY LEVELS AND WAVE FUNCTIONS. |
| Συγγραφείς: |
Бабенко, В. А.1 pet2@ukr.net |
| Πηγή: |
Nuclear Physics & Atomic Energy. 2026, Vol. 27 Issue 2, p125-141. 17p. |
| Θεματικοί όροι: |
*Anharmonic oscillator, *Rayleigh-Ritz method, *Energy levels (Quantum mechanics), *Quantum mechanics, *Numerical analysis, *Wave functions, *Variational approach (Mathematics) |
| Περίληψη: |
For the quantum quartic anharmonic oscillator with Hamiltonian H = 1/2 (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) = 1/2 (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 is shown that the proposed method provides standard 10−8 accuracy in energy calculations, employing a remarkably small basis size of N ≃ 6–7, while maintaining high efficiency throughout the strong-coupling region – for all values of λ. The behavior of the optimized frequency parameter ω0 with respect to the basis size N was also investigated, along with the identification and interpretation of the important effect of the variational plateau – a broad region of variation in the parameter ω0 where the computed energy remains nearly constant. The system’s wave functions were also calculated using the proposed method for a range of λ. The obtained results demonstrate that nonlinear optimization of the basis is a highly effective means of substantially increasing the efficiency of variational calculations, both for energies and for the wave functions. [ABSTRACT FROM AUTHOR] |
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