Exact Solutions And Numerical Calculation Of Typical Many-body Models

Exactly solved quantum many-body models are the basis of studying quan-turn many body problems and dealing with many complicated problems with numerical method. Although it has been developed over a hundred years, the number of exactly solved quantum many-body models is very few. The main reason is that there are mutual coupling in quantum many-body problems, which made it very difficult to exactly solve them. So taking some methods to decouple them becomes the essential element to deal with these problems. People usually use the following methods to decouple them:coordinate transformation, second quantization, mathematical physics method, as well as induction and deduction, or guess and construct the form of wavefunction and then proving them, and so on. Sometimes it needs several these methods meanwhile to solve the quantum many-body issue.In this thesis, we mainly study the solution of two interaction models and their ground-state properties:One model is one dimensional magnetized TG gas in an external magnetic field, and the other is quadratic pair potential model. The main content of this thesis is as follows:In chapter one, we briefly introduce the developed progress and current status of the quantum many-body models.In chapter two, in the first two sections, we state the general process of solving the hard-core Bose system with the famous Bose-Fermi correspondence principle, and then briefly introduce several relevant TG gas model and their main conclusions. After that, we introduce the methods of physicists often use to solve quadratic pair potential models and its conclusions. In the last section, we give definitions of some physical quantity which we pay attention to.In chapter three, we mainly study magnetized hard core Bose gas in an exter-nal periodic magnetic field in one dimension. Through the study of the TG gas, we obtain the following conclusions:First, when the number of bosons N is com-mensurate with the number of potential cycles M, it is easier to realize the ground state of TG gas while the system is in an external periodic potential. Second, TG gas and the corresponding spinless Fermi gas show the same phenomenon when we only concern physical quantities that not reflect self-correlations, so it is difficult to distinguish them. But they show distinct properties in physical quantities that reflect self-correlations. TG gas exhibit the condensate properties while it should be as Boson system. In this sense, self-correlation is more fundamental correlation and could further reflects the physical nature of correlations. Moreover, in the T-G gas models, on the basis of feature that TG gas is a special Bose system, we have investigated the manifestation of Bose Einstein Condensate (BEC), which is only existed in Bose systems. And we achieve three equations which used to judge whether BEC is presented or not. Using these three equations, we induct three well-know criteria that judge whether BEC is presented or not. With these three equations, it is easy to find the existed result that there is no BEC in one dimensional TG gas. And we also apply the. three equations to the whole one dimensional periodic Bose systems.In chapter four, we mainly calculate the energy spectrum and eigenvectors of quadratic pair potential systems. Based on lots of quantities of exact solutions of fixed number system, we sum up and obtain one conclusion:All the states can be obtained by only changing the last quantum number nN-1when the system constituted by the first N-1particles (in other words, by the first N-2quasi-particles) is in ground state. With this conclusion and mathematical induction, we could obtain the ground state wavefunction and then get all the exact eigenvalues and the corresponding eigenfunctions of N-body identical particle systems in one dimension. We find that all the energy levels are non-degenerate and the original first excited state or energy level disappears because of the operation of symmetry or antisymmetry of identical particles. For two and higher dimensions, it is so difficult, that we just give the energy spectrum, the analytical ground state wave function and the degree of degeneracy by induction method (The result of Bose system is trivial and it is easy to obtain without calculate). In addition, we roughly calculate the size of the system and pair distribution functions. While the system is in three dimension and the value of the energy gap is about the energy of gamma photon, the size of the system is coincide with atomic nucleus.At last, we give the summary and outlook of this thesis.