| Bose-Einstein condensation (BEC) is a ubiquitous phenomenon relating to many fields in physics. In 1924, Einstein predicted the phenomenon of condensation for an ideal gas of neutral atoms obeyed Bose-statistics under the temperature below critical value. Up to 1995, this idea were demonstrated in experiment of near-perfect gas of alkali atom 87 Rb ,23 Ma and 7 Li by loading laser-cooled atoms into a magnetic trap and successive cooling by evaporating, the BEC of atomic hydrogen was observed in 1998. A million or so atoms in the macroscopic quantum system are all evolving in a coherent way. They have completely lost their individual identity. The quantum phenomenon is brought in macroscopic scale by BEC. The extensive studies on the properties of a new state of matter have rapid developments in recent years. Based on a lot of the references relating to the experiment and the theory of BEC we have done some theoretical work in the following respects. (1) An analytic solution of the energy eigenvalue and wave function of neutral atoms with repellent and attractive interactions in a box potential are obtained. Compared with the system without interactions, with increase of interactions or number of atom, the energy levels of repellent system increases and the distributions of atoms approach decentralization, whereas the energy levels of attractive system decreases and the distributions of atoms approach centeralization. (2) A general approach to the BEC of neutral atoms with interaction is presented. The critical temperature of imperfect gas BEC is estimated. With the increasing of interaction the number atom hold in exited states decreases and the critical temperature increases. (3) We present a numerical method to solve the stationary nonlinear Schrodinger equation (NLSE) with an external potential, by adjusting eigenvalue and boundary value of wave function ' (0), such that theconditions of convergence and normalization of the wave function being satisfied simultaneously. We discuss the questions of convergence andnormalization of NLSE wave function. The accuracy of calculation is analyzed. (4) The numerical solutions of neutral atoms with attractive interaction in a harmonic trap and a spherically symmetric well potential of depth infinite are calculated. And the characteristics of two solutions are analogous on the compressed atom spatial distributions and the bistability curves of condensate atoms versus the energy eigenvalues. The critical point divides the eigenstates into metastable condensate and dense state, and gives the maximum number of atoms with which an attractive BEC can hold, and which is in agreement with the experiment on the whole. Based on this we present an exactly solvable model for macroscopic quantum tunneling of BEC with attractive interaction, and calculate the rate of macroscopic quantum tunneling from a metastable condensate state to the collapse state and analyze the stability of the attractive BEC. (5)According to recent quantum kinetic theory and using the generating function method to solve the master equation of BEC, we evaluate the growth rate, statistical fluctuation of condensate atoms. It is obtained that there is a plateau in the growth rate curve and a super-Poissonian distribution of statistical fluctuation. It is observed that the growth rate decreases progressively withthe increase of the final equilibrium value of condensate neq, which agrees with the experimental result on the whole. (6) An exactly solved model for the emission by N atoms is presented, the spontaneous and induced transition rates are enhanced by a factor which is proportional to the number of atoms n in the volume 3/(2 2) ( is the transition wavelength of atom) and dependent on the de-Broglie wavelength /1B in a more complicated way. The enhancement factor ' attains its maximum y'=1.453 at B = 2 B / = 2.45 , then decays to zero for large 3 ; and the energy level Lamb shift factor ' approaches to -1.0 for large B .
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