The Application Of ATM Theory In One-dimensionalwave Mechanics

Several applications of the ATM method in one dimensional wave mechanics, including exact quantum condition, supersymmetrical quantum mechanics, quantum reflection, exact reflection and tunneling time are discussed in this paper.At the beginning of this paper, we discussed the theory of the ATM method briefly, including the similarity between optics and one dimensional wave mechanics, the deduction of the analytic transfer matrix and the basic phase equation. The essence of the ATM method is to replace an arbitrary continuous potential by a series of thin layers. While the number of these layers approaches to infinity and their widths tends to zero, this series of step-potential layers approaches the discussed potential. The wave function in each layer can be expressed by a linear combination of trigonometric functions. With the connecting conditions of the wave function and its derivative on the boundary, a transfer matrix can be easily obtained. Also an effective wave function is introduced into the derivation of phase equation, which helps to reduce these two connecting conditions to only one: the connecting condition of effective wave function. Consequently, a phase equation equivalent to the matrix eqution can be obtained and the effective wave function is the crucial link.A whole technology has been evolved to understand the solvable potential problems and even to discover new ones based on SUSY QM. The concept of a shape invariant potential (SIP) was introduced in 1983 and it soon turned out that all the standard explicitly solvable potentials belong to SIPs involving a translation of parameters. Inspired by SUSY QM, Comtet et al. proposed a modified WKB quantization condition (SWKB), which is found to give better accuracy than the WKB quantization condition for many problems and be exact for all SIPs. Exactness of the SWKB quantization condition for SIPs has been the subject of wide discussions. The ATM method successful explained why the SWKB quantization condition is exact for all the SIPs and we found a general rule which is obeyed by all the standard explicitly solvable potentials: the phase contribution of the scattered subwaves between the two turning points is invariant.Then an exact and analytical equation about transmission coefficient is derived in the tunneling problem. Using the concept of total wave vector, this equation from ATM method has a more compact form and more physical insight than others’. Its most significant advantage is that it is obtained without any mentions of turning points in this process. So it is able to be applied in a uniform treatment of both the cases whether the energy is higer or lower than the peak of barriers. Of course, latter one has a more simple expression, but they are the same in fact.In recent years, quantum reflection, which can occurs above a potential barrier or purely attractive potential, has drawn much attention, especially in area of cold atom. However the theories on this issue are very unclear. Previous theoretical work formulated by Friedrich and co-workers was based on the globally accurate wave functions constructed by matching the exact or highly accurate wave functions in the quantal regions to the WKB wave functions in the semiclassical regions and concentrated mainly on the long-range attractive potential tails. However, the application of the WKB theory is rather cumbersome and much restricted and since the globally accurate wave functions is difficult to obtain and the proposed physical picture is rather ambiguous. However, we derived from the exact equation about transmission coefficient a very simple formula, which can be applied to all the quantum reflection problems. And we pointed out that quantum reflection is nothing but the reflection of the scattered subwaves.Since numerous theoretical predictions seem to contradict each other, the tunneling time issue is replete with controversy. Among them, two principal frameworks are the Büttiker-Landauer traversal time and the Wigner-Eisenbud time the group delay or phase time. The recent advance in experimental techniques has made the direct measurements of this quantity increasingly accessible to laboratory experiments. There already exist various experiments done with“single photons,”light pulses, etc.; some of them even show apparent superluminality. However, these experiments have severed to add fuel to the current debate over tunneling time. Via the ATM method, we found the inner connection between the classical reflection time and the quantum reflection time, which is completely determined by the scattered subwaves. And we proposed exact tunneling time and reflection time expressions.