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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Institute for Nuclear Research, National Academy of Sciences of Ukraine
2025-03-01
|
| Series: | Ядерна фізика та енергетика |
| Subjects: | |
| Online Access: | https://jnpae.kinr.kyiv.ua/26.1/Articles_PDF/jnpae-2025-26-0005-Babenko.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849744151814340608 |
|---|---|
| author | V. A. Babenko A. V. Nesterov |
| author_facet | V. A. Babenko A. V. Nesterov |
| author_sort | V. A. Babenko |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-bfcbe1393f104be2ab8a4b3bc2c49eb4 |
| institution | DOAJ |
| issn | 1818-331X 2074-0565 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Institute for Nuclear Research, National Academy of Sciences of Ukraine |
| record_format | Article |
| series | Ядерна фізика та енергетика |
| spelling | doaj-art-bfcbe1393f104be2ab8a4b3bc2c49eb42025-08-20T03:22:41ZengInstitute for Nuclear Research, National Academy of Sciences of UkraineЯдерна фізика та енергетика1818-331X2074-05652025-03-01261524https://doi.org/10.15407/jnpae2025.01.005The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergenceV. A. Babenko0A. V. Nesterov1Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv, UkraineBogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kyiv, UkraineFor 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.https://jnpae.kinr.kyiv.ua/26.1/Articles_PDF/jnpae-2025-26-0005-Babenko.pdfanharmonic oscillatoroscillator basisquantum field theory. |
| spellingShingle | V. A. Babenko A. V. Nesterov The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergence Ядерна фізика та енергетика anharmonic oscillator oscillator basis quantum field theory. |
| title | The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergence |
| title_full | The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergence |
| title_fullStr | The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergence |
| title_full_unstemmed | The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergence |
| title_short | The quartic anharmonic oscillator – an oscillator-basis expansion approach. II. Study of the wave functions and acceleration of the expansions convergence |
| title_sort | quartic anharmonic oscillator an oscillator basis expansion approach ii study of the wave functions and acceleration of the expansions convergence |
| topic | anharmonic oscillator oscillator basis quantum field theory. |
| url | https://jnpae.kinr.kyiv.ua/26.1/Articles_PDF/jnpae-2025-26-0005-Babenko.pdf |
| work_keys_str_mv | AT vababenko thequarticanharmonicoscillatoranoscillatorbasisexpansionapproachiistudyofthewavefunctionsandaccelerationoftheexpansionsconvergence AT avnesterov thequarticanharmonicoscillatoranoscillatorbasisexpansionapproachiistudyofthewavefunctionsandaccelerationoftheexpansionsconvergence AT vababenko quarticanharmonicoscillatoranoscillatorbasisexpansionapproachiistudyofthewavefunctionsandaccelerationoftheexpansionsconvergence AT avnesterov quarticanharmonicoscillatoranoscillatorbasisexpansionapproachiistudyofthewavefunctionsandaccelerationoftheexpansionsconvergence |