Short-range Potential Role Of Quantum Chaos, The Two-particle System

When we research the chaotic character of conservative system in classical mechanics, The easiest and most effective method is the Poincare surface of section (POS) of the orbits in phase space which abide by the classical Hamiltonian. But because there is the uncertainty principle in quantum mechanics, the particle's momenta and coordinates can't be determined at the same time, we can't use the orbits in phase space to describe quantum chaos. In fact, according to Bohr corresponding theorem, the result achieved from which we apply quantum mechanics to macroscopic motion corresponds with that of the classical mechanics. So the chaotic character of dynamic system can be manifested in quantum property.Now there are three kinds of quantum phenomena which relate to the classical chaos: (1) Energy level statistics. It is has been known that the energy spectra statistic of a chaotic system agrees with Wigner distribution which is achieved from Random Matrix Theory and the one of a integrable system is Poisson distribution achieved originally from the irregular spectra. When observing energy level of a chaotic system changing with one of the parameter, there are many avoided crossing, which possibly is an important sign of quantum chaos. (2) Morphology of stationary state eigenfunction. Because of the difficulty to calculate the wave function and transitive matrix element in high excited state, chaotic character of wave function is known a little. Now many pay attention to the scar of wave function; (3) The time evolutionary character of nonstationary state eigenfuction. Inorder to avoid the limit of the uncertainty relation, we will compare the classical and quantum in the ensemble kinesis-levels. There is a time scaleTE. in thequantum ensemble distribute corresponding to the chaoticcharacter of the ensemble distribute in classical phase space, if t > tE , thequantum effect will restrict the increase of the degree of ensemble distribute irregularity. On the basis of the Gutzwiller trace formula, the semiclassical theory of the chaotic system is developed successfully. The theory succeeded in atomic physics and recently is taken into the condensed physics.In this paper, we have studied a system including two particles. Comparing to the single-particle system, in N-body problems, not only the external condition is considered, but also the mutual interaction, so it is more complex to compute than in the single-particle system. In my thesis, I studied a simpler two-particle system; it is composed of two spinless one-dimensional (1D) particles of masses m₁ = m₂=m₂ . We have studied two typeof perfect potentials (a Dirac interaction and nonzero distance interaction) existing between the two particles. The spectrum statistics and the corresponding classical system action have been given. When the potential is interaction, because of the symmetry of the Hamiltonian, the Poincare surface of section (POS) of the orbits in the classical phase space is integrable, which is corresponding to the Poisson distribution in the quantum manifestation, our compute accords with this distribution. The result indicates that in this case the system is integrable not only in classical but in quantum. When there is nonzero distance interaction, the classical ispseudointegrable, but the spectrum statistics transfer from Poission distribution to GOE distribution, that mean quantum system is non-integrable. The result does not accord with the corresponding relation between the classical and the quantum. In quantum mechanics, because the particle has wave-corpuscle duality, the particle can tunnel the potential barrier which is higher than its energy. In the case of 8 interaction potential, because the range of action is very small, the influence that the energy levels are suffered by the tunneling effect is smaller than that of the nonzero distance interaction; the system can't manifest the chaotic character. The conclusion shows that the wave-corpuscle duality and the tunneling...