Abstract
Given the Thomas–Fermi equation \(\sqrt{x}\varphi ''=\varphi ^{3 \over 2}\), this paper changes first the dependent variable by defining \(y(x) \equiv \sqrt{x \varphi (x)}\). The boundary conditions require that y(x) must vanish at the origin as \(\sqrt{x}\), whereas it has a fall-off behaviour at infinity proportional to the power \({1 \over 2}(1-\chi )\) of the independent variable x, \(\chi \) being a positive number. Such boundary conditions lead to a 1-parameter family of approximate solutions in the form \(\sqrt{x}\) times a ratio of finite linear combinations of integer and half-odd powers of x. If \(\chi \) is set equal to 3, in order to agree exactly with the asymptotic solution of Sommerfeld, explicit forms of the approximate solution are obtained for all values of x. They agree exactly with the Majorana solution at small x, and remain very close to the numerical solution for all values of x. Remarkably, without making any use of series, our approximate solutions achieve a smooth transition from small-x to large-x behaviour. Eventually, the generalized Thomas–Fermi equation that includes relativistic, non-extensive and thermal effects is studied, finding approximate solutions at small and large x for small or finite values of the physical parameters in this equation.
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Notes
If one looks for solutions of Eq. (1.2) in the form \(\varphi (x)=A x^{\alpha }\), one finds the consistency condition
$$\begin{aligned} {\alpha (\alpha -1)\over \sqrt{A}}=x^{(3+\alpha )\over 2}, \end{aligned}$$which implies that \(\alpha =-3\), and hence \({12 \over \sqrt{A}}=1\), which is solved by \(A=144\).
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Acknowledgements
The authors are grateful to the Dipartimento di Fisica “Ettore Pancini” of Federico II University for hospitality and support.
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Esposito, G., Esposito, S. A new approach to the Thomas–Fermi boundary-value problem. Eur. Phys. J. Plus 135, 491 (2020). https://doi.org/10.1140/epjp/s13360-020-00507-4
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DOI: https://doi.org/10.1140/epjp/s13360-020-00507-4