scholarly journals Exactly and quasi-exactly solvable ‘discrete’ quantum mechanics

2011 ◽  
Vol 369 (1939) ◽  
pp. 1301-1318 ◽  
Author(s):  
Ryu Sasaki
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Essential Role ◽  
Dynamical Symmetry ◽  
Oscillator Algebra ◽  
Discrete Variable ◽  
Shape Invariance ◽  
One Dimensional ◽  
Exactly Solvable ◽  
Creation Operators

A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q -oscillator algebra and the Askey–Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional ‘discrete’ quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.

scholarly journals RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM

2011 ◽  
Vol 26 (32) ◽  
pp. 5337-5347 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Laguerre Polynomials ◽  
Supersymmetric Quantum Mechanics ◽  
Shape Invariance ◽  
Order Analysis ◽  
First Order ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials ◽  
Differential Relations

A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.

Generalized Electromagnetic Fields Associated with the Hydrogen-Like Atom Problem

2018 ◽  
Vol 74 (1) ◽  
pp. 43-50 ◽  
Author(s):  
S.A. Bruce ◽  
J.F. Diaz-Valdes
Keyword(s):  
Quantum Mechanics ◽  
Electromagnetic Coupling ◽  
Relativistic Quantum ◽  
Scalar Potential ◽  
Symmetry Algebra ◽  
Dynamical Symmetry ◽  
Canonical Commutation Relation ◽  
Quantum Systems ◽  
Minimal Coupling ◽  
Exactly Solvable

AbstractIt is known that the principle of minimal coupling in quantum mechanics determines a unique interaction form for a charged particle. By properly redefining the canonical commutation relation between (canonical) conjugate components of position and momentum of the particle, e.g. an electron, we restate the Dirac equation for the hydrogen-like atom problem incorporating a generalized minimal electromagnetic coupling. The corresponding interaction keeps the $1/\left|\mathbf{q}\right|$ dependence in both the scalar potential $V\left({\left|\mathbf{q}\right|}\right)$ and the vector potential $\mathbf{A}\left(\mathbf{q}\right)$ ($\left|{\mathbf{A}\left(\mathbf{q}\right)}\right|\sim 1/\left|\mathbf{q}\right|$). This problem turns out to be exactly solvable; moreover, the eigenstates and eigenvalues can be obtained in an elementary fashion. Some feasible models within this approach are discussed. Then we make a few remarks about the breaking of supersymmetry. Finally, we briefly comment on the possible Lie algebra (dynamical symmetry algebra) of these relativistic quantum systems.

WKB method for potentials unbounded from below

2016 ◽  
Vol 30 (03) ◽  
pp. 1650003 ◽  
Author(s):  
Aleksandar Demić ◽  
Vitomir Milanović ◽  
Jelena Radovanović ◽  
Milenko Musić
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Wave Functions ◽  
Wkb Method ◽  
Symmetric Potential ◽  
Overlap Integral ◽  
Entire Spectrum ◽  
One Dimensional ◽  
Model Based ◽  
Exactly Solvable

Bound states degenerated in energy (and differing in parity) may form in one-dimensional quantum mechanics if the potential is unbounded from below. We focus on symmetric potential and present quasi-exactly solvable (QES) model based on WKB method. The application of this method is limited on slow-changing potentials. We consider the overlap integral of WKB wave functions [Formula: see text] and [Formula: see text] which correspond to energies [Formula: see text] and [Formula: see text], and by setting [Formula: see text], we determine the type of spectrum depending on parameter [Formula: see text] which arises from this method. For finite value [Formula: see text], we show that the entire spectrum will consist of degenerated bound states.

Exactly solvable problems of quantum mechanics and their spectrum generating algebras: A review

Open Physics ◽  
2007 ◽  
Vol 5 (2) ◽  
Author(s):  
Constantin Rasinariu ◽  
Jeffry Mallow ◽  
Asim Gangopadhyaya
Keyword(s):  
Quantum Mechanics ◽  
Shape Invariance ◽  
Exactly Solvable ◽  
Solvable Problems ◽  
Made In

AbstractIn this review, we summarize the progress that has been made in connecting supersymmetry and spectrum generating algebras through the property of shape invariance. This monograph is designed to be used by our fellow researchers, by other interested physicists, and by students at the graduate and even undergraduate levels who would like a brief introduction to the field.

ALGEBRAIC APPROACH TO THE KLEIN–GORDON EQUATION WITH HYPERBOLIC SCARF POTENTIAL

2011 ◽  
Vol 20 (01) ◽  
pp. 55-61 ◽  
Author(s):  
SHISHAN DONG ◽  
SHI-HAI DONG ◽  
H. BAHLOULI ◽  
V. B. BEZERRA
Keyword(s):  
Quantum Mechanics ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Algebraic Approach ◽  
Supersymmetric Quantum Mechanics ◽  
Shape Invariance ◽  
One Dimensional ◽  
Scarf Potential ◽  
Klein Gordon ◽  
Klein Gordon Equation

Using the shape invariance approach we obtain exact solutions of one-dimensional Klein–Gordon equation with equal types of scalar and vector hyperbolic Scarf potentials. This is considered in the framework of supersymmetric quantum mechanics method.

scholarly journals NEW FINDINGS IN QUANTUM MECHANICS (PARTIAL ALGEBRAIZATION OF THE SPECTRAL PROBLEM)

1989 ◽  
Vol 04 (12) ◽  
pp. 2897-2952 ◽  
Author(s):  
M.A. SHIFMAN
Keyword(s):  
Dynamical Symmetry ◽  
Dimensional System ◽  
Graded Algebras ◽  
One Dimensional ◽  
New Class ◽  
Hidden Symmetry ◽  
New Findings ◽  
Exactly Solvable ◽  
Interesting Development ◽  
Solvable Problems

We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (like the famous harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that an (arbitrary) part of the eigenvalues and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the Hamiltonian. For one-dimensional motion, this hidden symmetry is SU(2). The simplest one-dimensional system admitting algebraization for a part of the spectrum is the anharmonic oscillator with the x6 anharmonicity and a relation between the coefficients in front of x2 and x6. We review also more complicated cases with the emphasis on pedagogical aspects. The groups SU (2)× SU (2), SO(3) and SU(3) generate two-dimensional problems with the partial algebraization of the spectrum. Typically we get Schrödinger-type equations in curved space. An intriguing relation between the algebraic structure of the Hamiltonian and the geometry of the space emerges. Another interesting development is the use of the graded algebras which allow one to construct multi-component quasi-exactly-solvable Hamiltonians.

EXPANDING THE CLASS OF SPECTRAL PROBLEMS WITH PARTIAL ALGEBRAIZATION

1989 ◽  
Vol 04 (13) ◽  
pp. 3311-3318 ◽  
Author(s):  
M.A. SHIFMAN
Keyword(s):  
Dynamical Symmetry ◽  
One Dimensional ◽  
Spectral Problems ◽  
Hidden Symmetry ◽  
Exactly Solvable ◽  
Solvable Problems

The phenomenon of the partial algebraization in the Schrödinger-type equations with a hidden dynamical symmetry is discussed. A generalization of the Turbiner procedure is proposed which can expand the class of the quasi-exactly-solvable problems. A specific example relevant to a one-dimensional Hamiltonian with the SU(2) hidden symmetry is constructed.

scholarly journals HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS

2011 ◽  
Vol 26 (25) ◽  
pp. 1843-1852 ◽  
Author(s):  
C. QUESNE
Keyword(s):  
Quantum Mechanics ◽  
Orthogonal Polynomials ◽  
Higher Order ◽  
Wave Functions ◽  
Supersymmetric Quantum Mechanics ◽  
Solvable Potentials ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Exceptional Orthogonal Polynomials

Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

scholarly journals RICCATI EQUATION, FACTORIZATION METHOD AND SHAPE INVARIANCE

2000 ◽  
Vol 12 (10) ◽  
pp. 1279-1304 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
ARTURO RAMOS
Keyword(s):  
Quantum Mechanics ◽  
Riccati Equation ◽  
Factorization Method ◽  
Shape Invariance ◽  
One Dimensional ◽  
Shape Invariant ◽  
Basic Concepts

The basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed. The relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the solutions of Infeld and Hull are generalized in a simple way.

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