Abstract

This paper deals with Thomas-Fermi equation which is formulated as an Euler-Lagrange equation associated with the Fermi energy functional. Drawing upon advanced ingredients of Sobolev spaces and weak solutions, an analytic methodology is presented for the quantum correction near the origin of Thomas-Fermi equation. By this approach the existence and uniqueness of the minimizer for the energy functional of the Thomas-Fermi equation has been proved. It has been demonstrated that by the definition of such a functional and the relevant Sobolev spaces, the Thomas-Fermi equation, particularly of a neutral atom, extends to the nonlinear Poisson equation. Accordingly, weak solutions for more general Euler-Lagrange equation with more singularities are proposed.

Keywords

References

[1] E. H. Leib, A brief review of Thomas–Fermi theory, Princeton University, 2000.
[2] E. H. Leib, and B. Simon, ―The Thomas–Fermi theory of atoms, molecules and solids‖, Advances in Mathematics, 23, pp. 22–116, 1977.
[3] F. Schwabl, Quantum mechanics, Springer , 2007.
[4] A. Hasan-Zadeh, and H. Fatoorehchi, ―Some analytic efforts for the quantum effects near the singularity of Thomas-Fermi equation‖, 1th National Conference on Applied Researches in Science and Engineering, Mashhad, and Islamic World Science Citation Center, 2017.
[5] H. Fatoorehchi, and H. Abolghasemi, ―An explicit analytic solution to the Thomas Fermi equation by improved differential transform method‖, Acta Physica Polonica A, 125, pp. 1083–1087, 2014.
[6] S. Liao, ―An explicit analytic solution to the Thomas–Fermi equation‖, Applied Mathematics and Computation, 144, pp. 495–506, 2003.
[7] R. A. Pearson, and S. M. Richardson, Computational analysis of polymer processing, Applied Science Publishers LTD, 1983.
[8] H. Kliener, Path integrals in quantum mechanics, statistics, polymer physics, and finanical markets, World Scientific, 5th edition, 2009.
[9] B. E. Baaquie, ―A path integral approach to optic princing with stochastic volatility: some exact results‖, J. de Physique I, 7, pp. 17– 33, 1997.
[10] D. Bahuguna, V. Raghavendra, and B. V. Rathish Kumar, Topics in Sobolev Spaces and Applications, Alpha Science, 2002.
[11] V. Maz’ya, ―Sobolev space in Mathematics II: Applications in analysis and partial differential equation‖, International Mathematical Series, vol. 9, Tamara Rozhkovslaya Publisher, 2009.