oschl-Teller Potential Research Articles

Systematic speedup of path integrals of a genericN-fold discretized theory

We present and discuss a detailed derivation of a new analytical method that systematically improves the convergence of path integrals of a generic $N$-fold discretized theory. We develop an explicit procedure for calculating a set of effective actions $S^{(p)}$, for $p=1,2,3,...$ which have the property that they lead to the same continuum amplitudes as the starting action, but that converge to that continuum limit ever faster. Discretized amplitudes calculated using the $p$ level effective action differ from the continuum limit by a term of order $1/N^p$. We obtain explicit expressions for the effective actions for levels $p\le 9$. We end by analyzing the speedup of Monte Carlo simulations of two different models: an anharmonic oscillator with quartic coupling and a particle in a modified P\"oschl-Teller potential.

Group approach to the quantization of the Pöschl–Teller dynamics

The quantum dynamics of a particle in the Modified P\"oschl-Teller potential is derived from the group $SL(2,R)$ by applying a Group Approach to Quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the $SL(2,R)$ group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU(2) where all the states are physical.

New two-dimensional quantum models partially solvable by the supersymmetrical approach

New solutions for second-order intertwining relations in two-dimensional SUSY QM are found via the repeated use of the first order supersymmetrical transformations with intermediate constant unitary rotation. Potentials obtained by this method - two-dimensional generalized P\"oschl-Teller potentials - appear to be shape-invariant. The recently proposed method of $SUSY-$separation of variables is implemented to obtain a part of their spectra, including the ground state. Explicit expressions for energy eigenvalues and corresponding normalizable eigenfunctions are given in analytic form. Intertwining relations of higher orders are discussed.

Comment on “Origin of light-induced states in intense laser fields and their observability in photoelectron spectra”

We report discrepancies between the results presented in Fig. 1 of a recent paper of Yasuike and Someda [Phys. Rev. A66, 053410 (2002)] and our independent calculation. At the frequency $\ensuremath{\omega}=0.55\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$, we find that the state of the one-dimensional modified P\oschl-Teller potential, described by the authors as light induced and originating from a shadow of the field-free ground state, is in fact physical for ${\ensuremath{\alpha}}_{0}l10\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and its origin is the zero-energy antibound state of the bare potential. For $\ensuremath{\omega}=0.45\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$, we also find differences in one of the presented quasienergy trajectories in the low ${\ensuremath{\alpha}}_{0}$ region $({\ensuremath{\alpha}}_{0}l0.4\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.})$, but we confirm the starting point at $E=\ensuremath{-}0.5\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ for both quasienergies, as found by Yasuike and Someda.

Supersymmetry and algebraic Darboux transformations

We describe a class of algebraically solvable SUSY models by considering the deformation of invariant polynomial flags by means of the Darboux transformation. The algebraic deformations corresponding to the addition of a bound state to a shape-invariant potential are particularly interesting. The polynomial flags in question are indexed by a deformation parameter m=1,2,... and lead to new algebraically solvable models. We illustrate these ideas by considering deformations of the hyperbolic P\"oschl-Teller potential.

Effective-Mass Schrödinger Equation and Generation of Solvable Potentials

A one-dimensional Schr\"odinger equation with position-dependent effective mass in the kinetic energy operator is studied in the framework of an $so(2,1)$ algebra. New mass-deformed versions of Scarf II, Morse and generalized P\"oschl-Teller potentials are obtained. Consistency with an intertwining condition is pointed out.

Quantum-information entropies of the eigenstates and the coherent state of the Pöschl-Teller potential

The position and momentum space information entropies, of the ground state of the P\"oschl-Teller potential, are exactly evaluated and are found to satisfy the bound, obtained by Beckner, Bialynicki-Birula and Mycielski. These entropies for the first excited state, for different strengths of the potential well, are then numerically obtained. Interesting features of the entropy densities, owing their origin to the excited nature of the wave functions, are graphically demonstrated. We then compute the position space entropies of the coherent state of the P\"oschl-Teller potential, which is known to show revival and fractional revival. Time evolution of the coherent state reveals many interesting patterns in the space-time flow of information entropy.

Integration of the Schrödinger equation by canonical transformations

Owing to the operator nature of the quantum dynamical variables, classical canonical transformations for integrating the equations of motion cannot be extended to the quantum domain. In this paper, a general procedure is developed to construct the sequences of quantum canonical transformations for integrating the Schr\"odinger equations. The sequence is made of three elementary canonical transformations that constitute a much larger class than the unitary transformations. In conjunction with the procedure, we also developed a factorization technique that is analogous to the method of integration factor in classical integration. For demonstration, with the same procedure we integrate nine nontrivial models, including the centripetal barrier potential, the Kratzer's molecular potential, the Morse potential, the P\"oschl-Teller potential, the Hulth\'en potential, etc.

Generalized intelligent states for an arbitrary quantum system

Generalized Intelligent States (coherent and squeezed states) are derived for an arbitrary quantum system by using the minimization of the so-called Robertson-Schr\"odinger uncertainty relation. The Fock-Bargmann representation is also considered. As a direct illustration of our construction, the P\"oschl-Teller potentials of trigonometric type will be shosen. We will show the advantage of the Fock-Bargmann representation in obtaining the generalized intelligent states in an analytical way. Many properties of these states are studied.

Mapping Wigner distribution functions into semiclassical distribution functions

A mapping that relates the Wigner phase-space distribution function of a given stationary quantum mechanical wave function, a solution of the Schr\"odinger equation, to a specific solution of the Liouville equation, both subject to the same potential, is studied. By making this mapping, bound states are described by semiclassical distribution functions still depending on Planck's constant, whereas for elastic scattering of a particle by a potential they do not depend on it, the classical limit being reached in this case. Following this method, the mapped distributions of a particle bound in the P\"oschl-Teller potential and also in a modified oscillator potential are obtained.

Features of time-independent Wigner functions

The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.

Generalized coherent states for the Pöschl-Teller potential and a classical limit

Coherent states in the harmonic oscillator may be defined in several equivalent ways. One definition describes coherent states as special states satisfying a minimum-uncertainty requirement in position and momentum spaces. This definition is generalized for other potentials according to a method developed by Nieto et al. [Phys. Rev. D 20, 1321 (1979)] and is herein applied to the P\"oschl-Teller potential. A classical limit based on the harmonic-oscillator coherent-state classical limit is then developed and applied to the P\"oschl-Teller minimum-uncertainty states. In this limit, classical behavior may be obtained from these quantum states. Together with a completeness argument for the generalized coherent states, this result provides insight into quantum classical correspondence through the statistical interpretation of quantum mechanics.

Modified Riccati approach to partially solvable quantum Hamiltonians. III. Related families of Pöschl-Teller potentials.

Within the modified Riccati approach developed elsewhere, we construct quasiexactly solvable potentials related to the P\"oschl-Teller and modified P\"oschl-Teller potentials. This is achieved by considering closed-form wave functions whose logarithmic derivatives are rational in the mapping u(x)=cosh(x) or u(x)=cos(x). With the hyperbolic mapping we construct an interesting four-parameter quasiexactly solvable confining potential with one, two, or three minima upon the value of its coupling constants. We give explicit expressions for the bound-state eigenfunctions and corresponding energy levels for some particular cases. The analytic continuation x\ensuremath{\rightarrow}ix transforms the previous potential into a periodic confining well, or a Kronig-Penney-like potential with an interesting structure of minima in each cell and a finite number of gaps. The energy levels in the last case are some of the band edges of the spectrum. In any case a finite but arbitrarily large number of energy levels and corresponding eigenfunctions are determined from the diagonalization of a finite matrix.

Exactly solvable potentials and quantum algebras.

A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials. General solution includes Shabat's infinite number soliton system and leads to raising and lowering operators satisfying $q$-deformed harmonic oscillator algebra. In the latter case energy spectrum is purely exponential and physical states form a reducible representation of the quantum conformal algebra $su_q(1,1)$.

Generalized deformed oscillators for vibrational spectra of diatomic molecules.

Using deformed bosons, we construct generalized anharmonic oscillators suitable for the description of vibrational spectra of diatomic molecules. One of them gives the same spectrum as the modified P\oschl-Teller potential, i.e., the first two terms of the Dunham expansion, while the other is a q deformation of the first, corresponding to summing all terms of the Dunham expansion for vibrational molecular spectra.

Classical potentials for q-deformed anharmonic oscillators.

Classical potentials giving the same spectrum as the q-deformed anharmonic oscillators having the symmetries ${\mathrm{U}}_{\mathit{q}}$(2)\ensuremath{\supset}O(2) and ${\mathrm{SU}}_{\mathit{q}}$(1,1), which have been used for the description of vibrational spectra of diatomic molecules, are determined in analytic form. The potentials found here are somewhat wider than the corresponding classical anharmonic potentials obtained in the limit q\ensuremath{\rightarrow}1, while their Taylor expansions are similar to the expansion of the modified P\"oschl-Teller potential, which is connected to the Morse potential through a known transformation.

Tunneling near the peaks of potential barriers: Consequences of higher-order Wentzel-Kramers-Brillouin corrections.

We derive the quantum-mechanical tunneling transmission coefficient for energies near the peak of a general potential barrier, to fourth order in the Wentzel-Kramers-Brillouin (WKB) approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the P\"oschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.

Path integrals over the SU(2) manifold and related potentials

A path-integral formalism is developed on the SU(2) manifold parametrized in terms of the Euler angles. The Green's function is studied for the P\oschl-Teller potential and the corresponding wave functions and energy spectrum are obtained.

Algebraic Approach to the Scattering Matrix

A purely algebraic procedure is presented for calculating recursion relations for $S$ matrices belonging to problems associated with the group SU(1,1). The procedure involves supplementing a recently introduced group-theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior. Our formulation makes use of the Euclidean group, which contains the symmetry transformations for a free particle. The Coulomb and P\"oschl-Teller potentials are discussed to illustrate our methods.

Lie Algebras for Potential Scattering

We consider a new Lie-algebraic framework for the P\"oschl-Teller potential, in which a single [$\mathrm{sp}(2,R)$ Casimir] operator has both bound and scattering states. This approach allows the determination of the $S$ matrix by purely algebraic means.