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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.17133 |
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| _version_ | 1866918129709350912 |
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| author | Irfan, Shaheen Ahmad, Zaki Akbar, Nosheen Mansoor, Minal Sumbul, Hussnain |
| author_facet | Irfan, Shaheen Ahmad, Zaki Akbar, Nosheen Mansoor, Minal Sumbul, Hussnain |
| contents | In this work, the energy eigenvalues are calculated for the quadratic ($\frac{g^2 x^2}{2}$), pure quartic ($λx^4 $), and quartic anharmonic oscillators ($\frac{g^2 x^2}{2} + λx^4 $) by applying variational method. For this, simple harmonic oscillator wave functions are considered as trial wave functions to calculate the energies for the ground state and first ten excited states with $g = 1$ and $λ=1/4$. For quartic anharmonic oscillators, energy values are calculated at different values of $λ$ with $g=1$. These energies for the ground state are compared with available numerically calculated data. Maximum value of $\%$error is found to be 1.9977. To get more accurate results, a new set of trial wave functions is suggested. With the newly proposed wave functions, maximum value of $\%$ error for the energy values reduces to 0.561. In this work, energies for the ground and first five excited states of quartic anharmonic oscillators are reported at different values of $λ$. Dependence of $λ$ on the wave functions is observed and concluded that wave functions are converging (shrinking) by increasing the $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17133 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy eigenvalues of quadratic, pure quartic and quartic anharmonic oscillators with variational method Irfan, Shaheen Ahmad, Zaki Akbar, Nosheen Mansoor, Minal Sumbul, Hussnain Quantum Physics In this work, the energy eigenvalues are calculated for the quadratic ($\frac{g^2 x^2}{2}$), pure quartic ($λx^4 $), and quartic anharmonic oscillators ($\frac{g^2 x^2}{2} + λx^4 $) by applying variational method. For this, simple harmonic oscillator wave functions are considered as trial wave functions to calculate the energies for the ground state and first ten excited states with $g = 1$ and $λ=1/4$. For quartic anharmonic oscillators, energy values are calculated at different values of $λ$ with $g=1$. These energies for the ground state are compared with available numerically calculated data. Maximum value of $\%$error is found to be 1.9977. To get more accurate results, a new set of trial wave functions is suggested. With the newly proposed wave functions, maximum value of $\%$ error for the energy values reduces to 0.561. In this work, energies for the ground and first five excited states of quartic anharmonic oscillators are reported at different values of $λ$. Dependence of $λ$ on the wave functions is observed and concluded that wave functions are converging (shrinking) by increasing the $λ$. |
| title | Energy eigenvalues of quadratic, pure quartic and quartic anharmonic oscillators with variational method |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2508.17133 |