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PATH INTEGRAL SOLUTIONS FOR DEFORMED P ÖSCHL–TELLER-LIKE AND CONDITIONALLY SOLVABLE POTENTIALS

Christian Grosche

Published Jan 1, 1999• 28 citations
OpenAlex

Abstract

I discuss in this paper the behaviour of the solutions of the so-called q-hyperbolic potentials, i.e. Pöschl–Teller-like and conditionally solvable potentials, in terms of the path integral formalism. The differences in comparison to the usual Pöschl–Teller-like potentials are investigated, including the discrete energy spectra and the bound state wave-functions. 1 Introduction. In this paper I want to discuss some specific generalisations of Pöschl–Teller related potentials. They are based on a q-deformation of the usual hyperbolic potentials, and are denoted by (we assume without loss of generality q> 0) Consequently we define

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I discuss in this paper the behaviour of the solutions of the so-called q-hyperbolic potentials, i.e. Pöschl–Teller-like and conditionally solvable potentials, in terms of the path integral formalism. The differences in comparison to the usual Pöschl–Teller-like potentials are investigated, including the discrete energy spectra and the bound state wave-functions. 1 Introduction. In this paper I want to discuss some specific generalisations of Pöschl–Teller related potentials. They are based on a q-deformation of the usual hyperbolic potentials, and are denoted by (we assume without loss of generality q> 0) Consequently we define
Abstract Under the condition of the spin symmetry, we rigorously solve the Dirac equation with the q − deformed hyperbolic Pöschl-Teller potential plus a trigonometric Scarf II potential. According to the values of the deformation parameter q, four different cases are considered. For the two cases q ≥ 1 and q ≤ − 1 with 1 2 α ln q < r < + ∞ , the analytical energy spectra and the spinor wave functions associated with the l d − wave bound states are obtained using a suitable approximation to the centrifugal potential term. When 0 < q < 1 or −1 < q < 0, the spinor wave functions for the s − wave bound states are derived and we find that the quantization conditions are transcendental equations which can be solved numerically. The special case q = 0 is also discussed.
Abstract In this paper, we solve the Feynman Kernel for the q-deformed hyperbolic Scarf potential for any ℓ -states. We propose an accurate generalization of the Pekeris approximation of the centrifugal term adapted to deformed potentials which allows to transform our potential to a solvable shifted modified Pöschl-Teller one. Analytical expressions of the energy spectrum and the normalized ℓ -state eigenfunctions are derived from the Green function. We evaluate and compare our results to the literature. Furthermore, we apply our method in chemical physics to derive the spectrum of some diatomic molecules. We obtained good results compared to state-of-the-art competing analytical and numerical methods.
Towards the Classification of Exactly Solvable Feynman Path Integrals: δδδ-Function Perturbations and Boundary-Problems as Miscellaneous Solvable Models
Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) Pöschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals.
In this paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half-plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half-plane to the existing one. All the calculations are mainly based on Fourier-expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Pöschl-Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave-functions and the energy-spectrum are derived. The energy-spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p -momentum). The “zero-point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.