Some exactly solvable bound-state problems

2012 ◽  
pp. 137-173
Author(s):  
Bipin R. Desai
Keyword(s):  
Bound State ◽  
Exactly Solvable

scholarly journals Trapping Charge Carriers in Low-Dimensional Dirac Materials

2019 ◽  
Vol 18 (03n04) ◽  
pp. 1940001
Author(s):  
C. A. Downing ◽  
M. E. Portnoi
Keyword(s):  
Bound States ◽  
Solvable Model ◽  
Mass Term ◽  
Bound State ◽  
Charge Carriers ◽  
Exactly Solvable Model ◽  
Dirac Materials ◽  
Exactly Solvable ◽  
Low Dimensional ◽  
Dependent Mass

We consider the problem of confining the famously elusive Dirac-like quasiparticles, as found in some recently discovered low-dimensional systems. After briefly surveying the existing theoretical proposals for creating bound states in Dirac materials, we study relativistic excitations with a position-dependent mass term. With the aid of an exactly-solvable model, we show how bound states begin to emerge after a critical condition on the size of the mass term is met. We also reveal some exotic properties of the unusual confinement discovered, including an elegant chevron structure of the bound state energies as a function of the size of the mass.

An approach to heavy quarkonium spectra

2014 ◽  
Vol 29 (32) ◽  
pp. 1450170 ◽  
Author(s):  
Y. Cançelik ◽  
B. Gönül
Keyword(s):  
Eigenvalue Problem ◽  
Coulomb Potential ◽  
Mass Spectra ◽  
Bound State ◽  
Considerable Effect ◽  
Heavy Quarkonia ◽  
Heavy Quarkonium ◽  
Related Literature ◽  
Exactly Solvable ◽  
Math Phys

An application of the recently introduced method [M. Çapak et al., J. Math. Phys. 52, 102102 (2011)] to the bound-state eigenvalue problem in the elementary quarkonium potential V(r) = -a/r + br + cr2 is described, proved and illustrated for [Formula: see text] and [Formula: see text] systems. The quasi- and conditionally-exactly solvable spin-averaged mass spectra of heavy quarkonia are obtained in compact forms. The comparison of the present predictions with those of other theories in the related literature, together with the available data, has shown the success of the model used in this work and also revealed that the use of different confinings in the perturbed Coulomb potential descriptions has no considerable effect on the mass spectra of such systems.

scholarly journals RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS

2017 ◽  
Vol 57 (6) ◽  
pp. 477 ◽  
Author(s):  
Rajesh Kumar Yadav ◽  
Nisha Kumari ◽  
Avinash Khare ◽  
Bhabani Prasad Mandal
Keyword(s):  
Canonical Transformation ◽  
Bound State ◽  
Invariance Property ◽  
Shape Invariance ◽  
Point Canonical Transformation ◽  
Solvable Potentials ◽  
Shape Invariant ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of <em>X<sub>m</sub></em> Laguerre or <em>X<sub>m</sub></em> Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

scholarly journals AN ALTERNATIVE SIMPLE SOLUTION OF THE SEXTIC ANHARMONIC OSCILLATOR AND PERTURBED COULOMB PROBLEMS

2007 ◽  
Vol 18 (10) ◽  
pp. 1571-1581 ◽  
Author(s):  
SAMEER M. IKHDAIR ◽  
RAMAZAN SEVER
Keyword(s):  
Wave Function ◽  
Wave Equation ◽  
Exact Solutions ◽  
Anharmonic Oscillator ◽  
Bound State ◽  
Exact Bound ◽  
Coulomb Problem ◽  
Radial Schrödinger Equation ◽  
Exactly Solvable ◽  
Coupling Parameters

Utilizing an appropriate ansatz to the wave function, we reproduce the exact bound-state solutions of the radial Schrödinger equation to various exactly solvable sextic anharmonic oscillator and confining perturbed Coulomb models in D-dimensions. We show that the perturbed Coulomb problem with eigenvalue E can be transformed to a sextic anharmonic oscillator problem with eigenvalue [Formula: see text]. We also check the explicit relevance of these two related problems in higher-space dimensions. It is shown that exact solutions of these potentials exist when their coupling parameters with k = D +2ℓ appearing in the wave equation satisfy certain constraints.

scholarly journals EXACTLY SOLVABLE EFFECTIVE MASS D-DIMENSIONAL SCHRÖDINGER EQUATION FOR PSEUDOHARMONIC AND MODIFIED KRATZER PROBLEMS

2009 ◽  
Vol 20 (03) ◽  
pp. 361-372 ◽  
Author(s):  
SAMEER M. IKHDAIR ◽  
RAMAZAN SEVER
Keyword(s):  
Schrödinger Equation ◽  
Effective Mass ◽  
Schrodinger Equation ◽  
Arbitrary Dimension ◽  
Bound State ◽  
Exact Bound ◽  
Point Canonical Transformation ◽  
Energy Eigenvalues ◽  
Exactly Solvable ◽  
Molecular Potentials

The point canonical transformation (PCT) approach is used to solve the Schrödinger equation for an arbitrary dimension D with a power-law position-dependent effective mass (PDEM) distribution function for the pseudoharmonic and modified Kratzer (Mie-type) diatomic molecular potentials. In mapping the transformed exactly solvable D-dimensional (D ≥ 2) Schrödinger equation with constant mass into the effective mass equation by using a proper transformation, the exact bound state solutions including the energy eigenvalues and corresponding wave functions are derived. The well-known pseudoharmonic and modified Kratzer exact eigenstates of various dimensionality is manifested.

scholarly journals REVISITING (QUASI-)EXACTLY SOLVABLE RATIONAL EXTENSIONS OF THE MORSE POTENTIAL

2012 ◽  
Vol 27 (13) ◽  
pp. 1250073 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Morse Potential ◽  
Bound State ◽  
Invariance Property ◽  
Shape Invariance ◽  
Point Canonical Transformation ◽  
First Order ◽  
Shape Invariant ◽  
Exactly Solvable ◽  
Parameter Values ◽  
Morse Potentials

The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials VA, B, ext (x), obtained from a conventional Morse potential VA-1, B(x) by the addition of a bound state below the spectrum of the latter, is reobtained. More importantly, the existence of another family of extended potentials, strictly isospectral to VA+1, B(x), is pointed out for a well-chosen range of parameter values. Although not shape invariant, such extended potentials exhibit a kind of "enlarged" shape invariance property, in the sense that their partner, obtained by translating both the parameter A and the degree m of the polynomial arising in the denominator, belongs to the same family of extended potentials. The point canonical transformation connecting the radial oscillator to the Morse potential is also applied to exactly solvable rationally-extended radial oscillator potentials to build quasi-exactly solvable rationally-extended Morse ones.

scholarly journals BOUND STATE WAVE FUNCTIONS THROUGH THE QUANTUM HAMILTON–JACOBI FORMALISM

2004 ◽  
Vol 19 (19) ◽  
pp. 1457-1468 ◽  
Author(s):  
S. SREE RANJANI ◽  
K. G. GEOJO ◽  
A. K. KAPOOR ◽  
P. K. PANIGRAHI
Keyword(s):  
Wide Class ◽  
Bound State ◽  
Wave Functions ◽  
Singularity Structure ◽  
State Wave ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Jacobi Formalism ◽  
Momentum Function

The bound state wave functions for a wide class of exactly solvable potentials are found by utilizing the quantum Hamilton–Jacobi formalism of Leacock and Padgett. It is shown that, exploiting the singularity structure of the quantum momentum function, until now used only for obtaining the bound state energies, one can straightforwardly find both the eigenvalues and the corresponding eigenfunctions. After demonstrating the working of this approach through a few solvable examples, we consider Hamiltonians, which exhibit broken and unbroken phases of supersymmetry. The natural emergence of the eigenspectra and the wave functions, in both unbroken and the algebraically nontrivial broken phase, demonstrates the utility of this formalism.

Sign in / Sign up

Export Citation Format

Share Document