The potential role of short-range two-particle system in quantum chaos
|Keywords||Quantum chaos Particle system Interaction potential Short Classical chaos Statistical distribution Quantum level Phase space orbit Uncertainty Relation Quantum systems|
People to study the chaotic characteristics of the classic conservative, the most simple and effective method is to calculate the phase space orbit Poincare. In quantum mechanics, however, due to the uncertainty relation, the particle momentum and coordinates can not be determined, so the classical Poincare emulated not apply. In fact, according to Bohr's correspondence principle, the quantum mechanics applied to macroscopic motion on the results obtained, it should be consistent with the results of classical mechanics, mechanical systems and therefore chaotic characteristics, are bound in their quantum nature something. At present, it has been found that the quantum and classical chaos phenomena are mainly three types, ① level spectra statistical characteristics. A classical chaotic system to meet the statistical distribution of the quantum energy spectrum by the random matrix theory derived Winger distribution, while the spectra of integrable systems to meet the statistical distribution of the random Poisson distribution. In the classic study of chaotic systems with a parameter change quantum level, people observed a large number of \② stationary state wave function morphology. As computing highly excited state wave functions and transition matrix element of difficulty, people on classical chaos to bring the characteristics of the wave function, little research work, the most current research is on the wave function scars. ③ unsteady-state wave function of the time evolution. Quantum mechanics in order to protect against uncertainty relation to limitations in movement level ensemble of classical and quantum dynamics for comparison, get classical ensemble chaotic phase space distribution characteristics in the corresponding quantum ensemble there is a distribution of motion timescales T_E, when t> T_E when quantum effects makes the ensemble distribution uniformity growth is inhibited. In Guziweile (Gutzwiller) trace equations, based on the semi-classical theory of chaotic systems have been greatly developed. It is successfully applied to the atomic physics among its recent applications in condensed matter physics has also attracted a large number of scientists. This thesis work: ten Hebei Normal University Master degree thesis we study a two-particle quantum system. Compared with the single-particle system in a multi-particle system, not only to consider the external conditions, but also consider the interaction between particles, so the multi-particle systems than the single-particle system complexity. In this paper we study a relatively simple two-particle system, the system by one-dimensional infinite potential well in the mass m. = MZ two m's two spinless identical particles of composition. There are two particles is calculated ideal short-range interaction potential (representing interaction potential and non-zero distance interaction potential) situation, given the system's level spectra statistical distribution. When accounting for the potential effects of particles, due to the symmetry of the Hamiltonian system corresponding classical phase space orbit Poincare chart emerged integrable nature, which correspond to the energy levels in the quantum statistics on the distribution of the random spectrum The Poisson distribution, we obtain the basis vectors expansion method is in line with the distribution of energy levels. This shows that in this case, whether classical or quantum performance performance systems are integrable. When the interaction potential between particles interaction potential non-zero distance, the corresponding classical system performance, though not integrable, but did not appear chaotic, showing the characteristic pseudo-integrable. Computing systems by numerical methods quantum level, level spectra found by the Poisson distribution statistical distribution but a gradual transition to GOE distribution, there has been the phenomenon of quantum chaos, this result and the general circumstances of the correspondence between classical and quantum match. This shows that we generally believe that the classical integrable quantum integrable also point does not apply in some cases, the reason will be the result, and quantum mechanics particles have wave-particle duality of a great relationship, the volatility of the particles it to a height greater than its own energy through the potential barrier. In accounting for the potential role, although there ll.J barrier tunneling effect, but due to the presence of a small range interaction potential, the energy levels of quantum systems affected t dagger when non-zero distance interaction potential is much smaller, so it will not statistical distribution appears GOE. Through both cases calculations show that the particles in the quantum system with wave-particle duality and the tunneling effect, so the system level spectra of quantum statistical characteristics will exhibit behavior inconsistent with their corresponding classical nature, and this kind nature and many-particle systems in the form of the interaction potential relevant.