Techniques Of Supersymmetric Quantum Mechanics Research Articles

A supersymmetric approach to the problem of micro-bending attenuation in optical waveguides

Micro-bending is a well-known source of loss in optical waveguides. By treating the micro-bending as a stochastic process, the problem of loss mitigation can be modeled in terms of a Fokker–Planck equation. Given an initial refractive index profile, and taking micro-bending into account, we develop a formalism to derive a new refractive index profile which potentially results in less loss. Our formalism is based on applying the techniques of Supersymmetric Quantum Mechanics to a Fokker–Planck equation that is associated with a particular refractive index profile. We derive a non-linear differential equation, whose solutions determine whether an index profile can undergo a supersymmetric transformation that results in less loss. As an explicit example, we consider a monomial index profile. We show that there exists a range of values for the monomial exponent which results in the new index profile having less loss.

Quantization of Some Generalized Jaynes-Cummings Models in a Kerr-Like Medium

Based on fundamental principles of quantum mechanics, we present a method for a rigorous analytic construction of the spectra of the one- and two-photon Jaynes-Cummings models in a Kerr medium. To obtain an idea of the method, we consider a first generalized Jaynes-Cummings model with a real, linear superpotential using techniques of supersymmetric quantum mechanics. The Hamiltonian of this model is written as a combination of operators generating the underlying superalgebra whose elements are defined as differential matrix operators. Based on the formalism of supersymmetric quantum mechanics and the properties of sets of common observables, we derive solutions of the one- and two-photon models expressed in terms of confluent hypergeometric functions. Finally, using numerical analysis, we study the influence of the Kerr effect on the energy spectra of the physical system.

Study of odd-A nuclei with an energy-dependent $$\gamma $$-unstable harmonic-oscillator-like potential and deformation-dependent mass term

Investigation of the Bohr Hamiltonian with Bose–Fermi dynamical symmetry is considered for odd–even nuclei near the critical point of the spherical to deformed $$\gamma $$-unstable phase transition $$[E(5|2j+1)]$$. This evaluation is modified by the energy-dependent harmonic-oscillator-like potential for $$\beta $$ part of the Hamiltonian in the presence of the deformation-dependent mass term, the spin–orbit interaction and the degeneracy-breaking terms. The Hamiltonian equation is solved using techniques of supersymmetric quantum mechanics (SUSYQM). The excitation energy spectra, the electromagnetic transition strengths, quadrupole moments and dipole moments for this case are given for $$^{191}\hbox {Ir}$$.

SUSYQM in nuclear structure: Bohr Hamiltonian with mass depending on the deformation

A well known problem of the Bohr Hamiltonian for the description of nuclear collective motion is that the nuclear moment of inertia increases with deformation too fast. We show that this can be avoided by allowing the nuclear mass to depend on the deformation. The resulting Hamiltonian is solved exactly, using techniques of Supersymmetric Quantum Mechanics

Group contractions and conformal maps in nuclear structure models

Group contraction is a procedure in which a symmetry group is reduced into a group of lower symmetry in a certain limiting case. Examples are provided in the large boson mumber limit of the Interacting Boson Approximation (IBA) model by a) the contraction of the SU(3) algebra into the [R5]SO(3) algebra of the rigid rotator, consisting of the angular momentum operators forming SO(3), plus 5 mutually commuting quantities, the quadrupole operators, b) the contraction of the O(6) algebra into the [R5]SO(5) algebra of the ∞-unstable rotator. We show how contrac- tions can be used for constructing symmetry lines in the interior of the symmetry triangle of the IBA model. In mathematics, a conformal map is a function which preserves angles. We show how this procedure can be used in the framework of the Bohr Hamiltonian, leading to a Hamiltonian in a curved space, in which the mass depends on the nuclear deformation Ø, while it remains independent of the collective variable ∞ and the three Euler angles. This Hamiltonian is proved to be equivalent to that obtained using techniques of Supersymmetric Quantum Mechanics.

Confinement of Dirac electrons in graphene magnetic quantum dots

We characterize the confinement of massless Dirac electrons under axially symmetric magnetic fields in graphene, including zero energy modes and higher energy levels. In particular, we analyze in detail the Aharonov–Casher theorem, on the existence of zero modes produced by magnetic fields with finite flux in two dimensions. We apply techniques of supersymmetric quantum mechanics to determine the confined states by means of the quantum number j associated to isospin and angular momentum. We focus on magnetic fields, regular at the origin, whose asymptotic behaviour is , with α a real number. A confinement of infinite zero-energy modes and excited states is possible as long as . When the quantum dot is able to trap an infinite number of zero modes but no excited states, while for only a finite number of zero modes are confined.

New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization

We derive a one-step extension of the well known Swanson oscillator that describes a specific type of pseudo-Hermitian quadratic Hamiltonian connected to an extended harmonic oscillator model. Our analysis is based on the use of the techniques of supersymmetric quantum mechanics and addresses various representations of the ladder operators starting from a seed solution of the harmonic oscillator expressed in terms of a pseudo-Hermite polynomial. The role of the resulting chain of Hamiltonians related to similarity transformation is then exploited. In the second part we write down a two dimensional generalization of the Swanson Hamiltonian and establish superintegrability of such a system.

Bohr Hamiltonian with deformation-dependent mass

The Bohr Hamiltonian with a mass depending on the nuclear deformation is solved using the techniques of supersymmetric quantum mechanics (SUSYQM). Analytical expressions for spectra and wave functions are obtained. Spectra and B(E2) transition rates are calculated for more than 50 γ-unstable nuclei and more than 60 prolate deformed nuclei using the Davidson potential and the Kratzer potential. In addition to solving the long standing problem of the too rapid increase of the moment of inertia with deformation, the method reveals a conformal factor in the Bohr Hamiltonian, embedding the Bohr space in six dimensions.

SUSY partners for spin-1/2 systems in nonrelativistic limits

Abstract In this paper, we use some well-known techniques of Supersymmetric QuantumMechanics (SUSYQM) namely the factorization method and shape invariance, to generate new analytically solvable potentials from some interacting fermionic models in nonrelativistic limits. These systems are described by the ordinary and the harmonically trapped Schrödinger-Pauli particle models and the Dirac-Coulomb Hamiltonian, this latter being set in its nonrelativistic limits. The spectrum for each of these models is obtained in a simple and transparent way. We then generate new solvable potentials that describe interactions between electromagnetic field and matter, paying due attention to the subtleties inherent in the application of SUSY to higher dimensional problems. SUSY breaking problems related to the partner singularities are dicussed along with the paper.

Bohr Hamiltonian with a deformation-dependent mass term for the Kratzer potential

The Deformation Dependent Mass (DDM) Kratzer model is constructed by considering the Kratzer potential in a Bohr Hamiltonian, in which the mass is allowed to depend on the nuclear deformation, and solving it by using techniques of supersymmetric quantum mechanics (SUSYQM), involving a deformed shape invariance condition. Analytical expressions for spectra and wave functions are derived for separable potentials in the cases of gamma-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. Spectra and B(E2) transition rates are compared to experimental data. The dependence of the mass on the deformation, dictated by SUSYQM for the potential used, moderates the increase of the moment of inertia with deformation, removing a main drawback of the model.

Group contractions and conformal maps in nuclear structure

Group contraction is a procedure in which a symmetry group is reduced into a group of lower symmetry in a certain limiting case. Examples are provided in the large boson number limit of the Interacting Boson Approximation (IBA) model by a) the contraction of the SU(3) algebra into the [R5]SO(3) algebra of the rigid rotator, consisting of the angular momentum operators forming SO(3), plus 5 mutually commuting quantities, the quadrupole operators, b) the contraction of the 0(6) algebra into the [R5]SO(5) algebra of the γ-unstable rotator. We show how contractions can be used for constructing symmetry lines in the interior of the symmetry triangle of the IBA model.In mathematics, a conformal map is a function which preserves angles. We show how this procedure can be used in the framework of the Bohr Hamiltonian, leading to a Hamiltonian in a curved space, in which the mass depends on the nuclear deformation β, while it remains independent of the collective variable 7 and the three Euler angles. This Hamiltonian is proved to be equivalent to that obtained using techniques of Supersymmetric Quantum Mechanics.

Scattering cross-section resonance originating from a spectral singularity

Using techniques of supersymmetric quantum mechanics, scattering properties of Hermitian Hamiltonians, which are related to non-Hermitian ones by similarity transformations, are studied. We have found that the scattering matrix of the Hermitian Hamiltonian coincides with the phase factor of the non-unitary scattering matrix of the non-Hermitian Hamiltonian. The possible presence of a spectral singularity in a non-Hermitian Hamiltonian translates into a pronounced resonance in the scattering cross section of its Hermitian counterpart. This opens a way for detecting spectral singularities in scattering experiments; although a singular point is inaccessible for the Hermitian Hamiltonian, the Hamiltonian ‘feels’ the presence of the singularity if it is ‘close enough’. We also show that cross sections of the non-Hermitian Hamiltonian do not exhibit any resonance behavior and explain the resonance behavior of the Hermitian Hamiltonian cross section by the fact that the corresponding scattering matrix, up to a background scattering matrix, is a square root of the Breit–Wigner scattering matrix.

Bohr Hamiltonian with a deformation-dependent mass term for the Davidson potential

Analytical expressions for spectra and wave functions are derived for a Bohr Hamiltonian, describing the collective motion of deformed nuclei, in which the mass is allowed to depend on the nuclear deformation. Solutions are obtained for separable potentials consisting of a Davidson potential in the beta variable, in the cases of gamma-unstable nuclei, axially symmetric prolate deformed nuclei, and triaxial nuclei, implementing the usual approximations in each case. The solution, called the Deformation Dependent Mass (DDM) Davidson model, is achieved by using techniques of supersymmetric quantum mechanics (SUSYQM), involving a deformed shape invariance condition. Spectra and B(E2) transition rates are compared to experimental data. The dependence of the mass on the deformation, dictated by SUSYQM for the potential used, reduces the rate of increase of the moment of inertia with deformation, removing a main drawback of the model.

Bohr Hamiltonian with deformation-dependent mass term

The Bohr Hamiltonian describing the collective motion of atomic nuclei is modified by allowing the mass to depend on the nuclear deformation. Exact analytical expressions are derived for spectra and wave functions in the case of a γ-unstable Davidson potential, using techniques of supersymmetric quantum mechanics. Numerical results in the Xe–Ba region are discussed.

Analytically solvable PT-invariant periodic potentials

Associated Lamé potentials V( x)= a( a+1) msn 2( x, m)+ b( b+1) mcn 2( x, m)/dn 2( x, m) are used to construct complex, PT-invariant, periodic potentials using the anti-isospectral transformation x→ ix+ β, where β is any nonzero real number. These PT-invariant potentials are defined by V PT( x)≡− V( ix+ β), and have a different real period from V( x). They are analytically solvable potentials with a finite number of band gaps, when a and b are integers. Explicit expressions for the band edges of some of these potentials are given. For the special case of the complex potential V PT( x)=−2 msn 2( ix+ β, m), we also analytically obtain the dispersion relation. Additional new, solvable, complex, PT-invariant, periodic potentials are obtained by applying the techniques of supersymmetric quantum mechanics.

Conditionally exactly solvable potentials and supersymmetric transformations

A general procedure is presented to construct conditionally exactly solvable (CES) potentials using the techniques of supersymmetric quantum mechanics. The method is illustrated with potentials related to the harmonic oscillator problem. Besides recovering known results, new CES potentials are also obtained within the framework of this general approach. The conditions under which this method leads to CES potentials are also discussed.

Domain walls of gauged supergravity, M-branes, and algebraic curves

We provide an algebraic classification of all supersymmetric domain wall solutions of maximal gauged supergravity in four and seven dimensions, in the presence of non-trivial scalar fields in the coset SL(8,R)/SO(8) and SL(5,R)/SO(5) respectively. These solutions satisfy first-order equations, which can be obtained using the method of Bogomol'nyi. From an eleven-dimensional point of view they correspond to various continuous distributions of M2- and M5-branes. The Christoffel-Schwarz transformation and the uniformization of the associated algebraic curves are used in order to determine the Schrodinger potential for the scalar and graviton fluctuations on the corresponding backgrounds. In many cases we explicitly solve the Schrodinger problem by employing techniques of supersymmetric quantum mechanics. The analysis is parallel to the construction of domain walls of five-dimensional gauged supergravity, with scalar fields in the coset SL(6,R)/SO(6), using algebraic curves or continuous distributions of D3-branes in ten dimensions. In seven dimensions, in particular, our classification of domain walls is complete for the full scalar sector of gauged supergravity. We also discuss some general aspects of D-dimensional gravity coupled to scalar fields in the coset SL(N,R)/SO(N).

Phase-equivalent potentials from supersymmetry: analytical results for a Natanzon-class potential

Applying the techniques of supersymmetric quantum mechanics we determine closed algebraic expressions for potentials that are phase-equivalent with the generalized Ginocchio potential, which is a member of the rather general Natanzon-potential class. In particular, we discuss the elimination of bound states, the addition of one (or more) bound state at specific energies and also mention transformations that leave the spectrum unchanged. Our work represents the application of the abstract mathematical formalism developed recently for the modification of the spectrum of potentials without changing the S-matrix and the phase shifts. A new aspect of our work is that in addition to the new potential function, we give closed analytical expressions for the transformed Jost functions and bound-state wavefunctions. Furthermore, this work is the first example for generating phase-equivalent partners of a potential outside the relatively simple shape-invariant potential class.

New Phase Equivalent Partners to Eckart and Bargmann Potentials

Using techniques of supersymmetric quantum mechanics we derive phase equivalent partner potentials to Eckart and Bargmann potentials. It is found that the partner potentials pick up new singularities which are absent in the parent potentials.

Quantum mechanics of multi-prong potentials

We describe the bound state and scattering properties of a quantum mechanical particle in a scalar $N$-prong potential. Such a study is of special interest since these situations are intermediate between one and two dimensions. The energy levels for the special case of $N$ identical prongs exhibit an alternating pattern of non-degeneracy and $(N-1)$ fold degeneracy. It is shown that the techniques of supersymmetric quantum mechanics can be used to generate new solutions. Solutions for prongs of arbitrary lengths are developed. Discussions of tunneling in $N$-well potentials and of scattering for piecewise constant potentials are given. Since our treatment is for general values of $N$, the results can be studied in the large $N$ limit. A somewhat surprising result is that a free particle incident on an $N$-prong vertex undergoes continuously increased backscattering as the number of prongs is increased.