Study On The Exact Quantization Rule And The Quantum Time

With previous results given by the ATM method, three basic quantum issues, which are the exact quantization rule, the quantum reflection time, and the quantum tunneling time, are discussed in this paper.At the beginning, we give a brief introduction to the theoretical basis of some quantum semi-classical methods, which include WKB approximation, EBK approximation, NMI method; and the advances in study of the quantum time. Then the establishment process of the ATM transfer matrix for an arbitrary one-dimensional potential is given. The arbitrary potential is divided into a series of thin layers, each local potential can be considered as a constant potential and this series of step-potentials becomes the discussed potential when each layer's width tends to 0. The wave function in each layer can be expressed as a linear combination of exponent functions. With the connecting condition of the wave function and its derivation at each boundary point, the transfer matrix can be easily obtained.Based on the proposition of the effective attenuation coefficient, the ATM method allows us to deduce the exact quantization rule. An important result is that the phase loss at a turning point is exactly equal toπ, rather than the usual choice ofπ2 or the other values in WKB approximation and its modified versions. Furthermore, a new concept of subwaves is put forward which is the basis of the following work of this paper but is ignored in the conventional semi-classical schemes. Using this exact ATM quantization rule, the exact energy spectrum of ECSC potential is also given.There are many unsolved problems in the research of quantum time, which including quantum reflection time and quantum tunneling time, etc. In quantum mechanics, the physical quantities can be characterized by the corresponding operators; however, until now the operator of quantum time has not been exactly found. Consequently, the problem that how long one particle passes through a one-dimensional quantum barrier is unclear. Numerous theories have been proposed to define the quantum time, but their contradictory results have given nothing but severed to add fuel to the curiosity of understanding this issue, and recently the experiment of the directly measurement of the quantum tunneling time have attracted considerable interests. In this paper, the generalized formula of the quantum reflection time and the quantum tunneling time have been presented on the basis of the analytical transfer matrix method. It is found that its concept of the subwaves is extremely important in these two quantum dynamic processions: the subwaves determines the discrepancy between the quantum reflection time and the classical reflection time; the tunneling time is exactly the result of the subwaves, and the Hartman effect can be explanted by the phenomenon that the subwaves is gradually disappeared with the extending of the potential barrier.