Study On Some Basic Quantum Issues By Using The Analytical Transfer Matrix Method

With previous results given by the ATM method, three basic quantum issues, which are quantization rule, quantum reflection & quantum tunneling, and complex reflectionless potential, 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 supersymmetry theory. 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 ofπ/ 2 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 and non-shape invariance potentials are also given.Beyond the reach of classical mechanics, quantum mechanics gives rise to a great number of peculiar physical phenomena. Among them, quantum reflection refers to a particle is reflected when it moves in an attracted potential or through a classically allowed region without reaching a classical turning point. Quantum tunneling refers to the phenomena of a particle's ability to penetrate potential barriers. In recent decades, an upsurge of interest for experimental research on the cold atom mirror and atoms hologram, and for nano-electronic devices design based on quantum tunneling is abundant in the physical literatures. On the other hand, it still remains a challenge to find a general and exact theory for explaining these quantum phenomena. Using the reflection and tunneling probabilities formula given by the ATM method, a novel explanation for these phenomena is proposed that quantum reflection is nothing but the reflection of the scattering subwaves and scattering subwaves also plays a crucial role in quantum tunneling. Numerical results are highly consistent with the proposed conclusion and this work may provide guidance and motivation for new experiments.In addition, we point out that it is wrong to obtain reflectionless potentials by using superpotential which is in terms of the discontinuous function. Different from previous works, we propose that complex potentials can be used to construct reflectionless potentials. In particular, we study the robustness of these reflectionless potentials in details.