Abstract
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Abstract
In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.
References
- Baer M., 2018, findiff Software Package, https://github.com/maroba/ findiff google scholar
- Bender C. M., Wu T. T., 1969, Physical Review, 184, 1231 google scholar
- Bender C. M., Wu T. T., 1973, Physical Review D, 7, 1620 google scholar
- Flügge S., 1999, One-Body Problems without Spin. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 25-331, doi:10.1007/978-3-642-61995-3_2 google scholar
- Gaudreau P., Slevinsky R. M., Safouhi H., 2013, Annals of Physics, 337, 261 google scholar
- Gaudreau P. J., Slevinsky R. M., Safouhi H., 2015, Annals of Physics, 360, 520 google scholar
- Harris C. R., et al., 2020, Nature, 585, 357 google scholar
- Hioe F., Macmillen D., Montroll E., 1978, Physics Reports, 43, 305 google scholar
- Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 google scholar
- Turbiner A. V., 1981, Journal of Physics A: Mathematical and General, 14, 1641 google scholar
- Turbiner A. V., 2005, Letters in Mathematical Physics, 74, 169 google scholar
- Turbiner A. V., del Valle J. C., 2021, Journal of Physics A: Mathemat-ical and Theoretical, 54, 295204 google scholar
- Van Rossum G., Drake F. L., 2009, Python 3 Reference Manual. Cre-ateSpace, Scotts Valley, CA google scholar
- Virtanen P., et al., 2020, Nature Methods, 17, 261 google scholar
Details
| Primary Language | English |
|---|---|
| Subjects | General Physics |
| Journal Section | Research Articles |
| Authors | |
| Publication Date | December 21, 2023 |
| Submission Date | October 2, 2023 |
| Published in Issue | Year 2023 |