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Register a new userTime dependent rationally extended Pu007foschl-Teller potential and some of its properties
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We examine time dependent Schru007fodinger equation with oscillating boundary condition. More specifically, we use separation of variable technique to construct time dependent rationally extended Pu007foschl-Teller potential (whose solutions are given by in terms of X1 Jacobi exceptional orthogonal polynomials) and its supersymmetric partner, namely the Pu007foschl-Teller potential. We have obtained exact solutions of the Schru007fodinger equation with the above mentioned potentials subjected to some boundary conditions of the oscillating type. A number of physical quantities like the average energy, probability density, expectation values etc. have also been computed for both the systems and compared with each other.
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We analyze the one dimensional scattering produced by all variations of the Poschl-Teller potential, i.e., potential well, low and high barriers. We show that the Poschl-Teller well and low barrier potentials have no resonance poles, but an infinite number of simple poles along the imaginary axis corresponding to bound and antibound states. A quite different situation arises on the Poschl-Teller high barrier potential, which shows an infinite number of resonance poles and no other singularities. We have obtained the explicit form of their associated Gamow states. We have also constructed ladder operators connecting wave functions for bound and antibound states as well as for resonance states. Finally, using wave functions of Gamow and antibound states in the factorization method, we construct some examples of supersymmetric partners of the Poschl-Teller Hamiltonian.
This is the fourth in a series of papers on developing a formulation of quantum mechanics in non-inertial reference frames. This formulation is grounded in a class of unitary cocycle representations of what we have called the Galilean line group, the generalization of the Galilei group to include transformations amongst non-inertial reference frames. These representations show that in quantum mechanics, just as the case in classical mechanics, the transformations to accelerating reference frames give rise to fictitious forces. In previous work, we have shown that there exist representations of the Galilean line group that uphold the non-relativistic equivalence principle as well as representations that violate the equivalence principle. In these previous studies, the focus was on linear accelerations. In this paper, we undertake an extension of the formulation to include rotational accelarations. We show that the incorporation of rotational accelerations requires a class of emph{loop prolongations} of the Galilean line group and their unitary cocycle representations. We recover the centrifugal and Coriolis force effects from these loop representations. Loops are more general than groups in that their multiplication law need not be associative. Hence, our broad theoretical claim is that a Galilean quantum theory that holds in arbitrary non-inertial reference frames requires going beyond groups and group representations, the well-stablished framework for implementing symmetry transformations in quantum mechanics.
The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forreys transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.
Nonextensive statistical mechanics has been a source of investigation in mathematical structures such as deformed algebraic structures. In this work, we present some consequences of $q$-operations on the construction of $q$-numbers for all numerical sets. Based on such a construction, we present a new product that distributes over the $q$-sum. Finally, we present different patterns of $q$-Pascals triangles, based on $q$-sum, whose elements are $q$-numbers.
We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schrodinger equation recently advanced by Nobre, Rego-Montero and Tsallis (NRT) [Phys. Rev. Lett. 106 (2011) 10601]. The NRT non-linear equation admits plane wave-like solutions ($q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the $q$-generalized thermostatistical formalism, is characterized by a parameter $q$, and in the limit $q to 1$ reduces to the standard, linear Schrodinger equation. The $q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known $q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the $q to 1$ limit the $q$-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schrodinger equation. In the present work we also show that there are other families of nonlinear Schrodinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.
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