Abstract
The Gross-Pitaevskii equation (GPE), that describes the wave function of a number of coherent Bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GPE consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a spectral method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained. This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.
Highlights
The phenomenon of Bose-Einstein condensation of an assembly of atoms, predicted in 1924 [1,2,3], was observed experimentally in 1995 [4,5] for atoms confined in a trap at very low temperatures
A method implemented here for the numerical solution of the Gross-Pitaevskii equation (GPE) consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential
A method is presented of solving for the L 0 partial wave function of the the Gross Pitaevskii nonlinear differential equation
Summary
The phenomenon of Bose-Einstein condensation of an assembly of atoms, predicted in 1924 [1,2,3], was observed experimentally in 1995 [4,5] for atoms confined in a trap at very low temperatures. This is a Schrödinger-like equation, called the Gross-Pitaevskii equation (GPE), describes the wave function of N Bose particles interacting coherently and confined in an atomic trap. In this equation only the short range part of the interaction between the atoms is included in terms of the scattering length of two colliding atoms. One aspect emphasized in the present study is the description of the coherent interaction of the atoms in the BEC in terms of a related Hartree potential, VH This potential arises naturally in the GPE, due to the presence of the third power of the wave function in that equation, by rewriting the term 3 as VH.
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