Path integral for the quartic oscillator: An accurate analytic formula for the partition function
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910612997537792 |
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| author | Caffarel, Michel |
| author_facet | Caffarel, Michel |
| contents | In this work an approximate analytic expression for the quantum partition function of the quartic oscillator described by the potential $V(x) = \frac{1}{2} ω^2 x^2 + g x^4$ is presented. Using a path integral formalism, the exact partition function is approximated by the partition function of a harmonic oscillator with an effective frequency depending both on the temperature and coupling constant $g$. By invoking a Principle of Minimal Sensitivity (PMS) of the path integral to the effective frequency, we derive a mathematically well-defined analytic formula for the partition function. Quite remarkably, the formula reproduces qualitatively and quantitatively the key features of the exact partition function. The free energy is accurate to a few percent over the entire range of temperatures and coupling strengths $g$. Both the harmonic ($g\rightarrow 0$) and classical (high-temperature) limits are exactly recovered. The divergence of the power series of the ground-state energy at weak coupling, characterized by a factorial growth of the perturbational energies, is reproduced as well as the functional form of the strong-coupling expansion along with accurate coefficients. Explicit accurate expressions for the ground- and first-excited state energies, $E_0(g)$ and $E_1(g)$ are also presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_09859 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Path integral for the quartic oscillator: An accurate analytic formula for the partition function Caffarel, Michel Quantum Physics Statistical Mechanics Atomic Physics Chemical Physics In this work an approximate analytic expression for the quantum partition function of the quartic oscillator described by the potential $V(x) = \frac{1}{2} ω^2 x^2 + g x^4$ is presented. Using a path integral formalism, the exact partition function is approximated by the partition function of a harmonic oscillator with an effective frequency depending both on the temperature and coupling constant $g$. By invoking a Principle of Minimal Sensitivity (PMS) of the path integral to the effective frequency, we derive a mathematically well-defined analytic formula for the partition function. Quite remarkably, the formula reproduces qualitatively and quantitatively the key features of the exact partition function. The free energy is accurate to a few percent over the entire range of temperatures and coupling strengths $g$. Both the harmonic ($g\rightarrow 0$) and classical (high-temperature) limits are exactly recovered. The divergence of the power series of the ground-state energy at weak coupling, characterized by a factorial growth of the perturbational energies, is reproduced as well as the functional form of the strong-coupling expansion along with accurate coefficients. Explicit accurate expressions for the ground- and first-excited state energies, $E_0(g)$ and $E_1(g)$ are also presented. |
| title | Path integral for the quartic oscillator: An accurate analytic formula for the partition function |
| topic | Quantum Physics Statistical Mechanics Atomic Physics Chemical Physics |
| url | https://arxiv.org/abs/2312.09859 |