Supersymmetry and Solution of Dirac Equation with Vector and Scalar Potentials

The Dirac equations with vector and scalar potentials of the Coulomb types in two and three dimensions are solved using the supersymmetric quantum mechanics method. For the system of such potentials, the analytical expressions of the matrix dements for both position and momentum operators are obtained.     

Dirac Equation and Some Quasi-Exact Solvable Potentials in the Turbiner's Classification

In this paper quasi-exact solvability(QES)of Dirac equation with some scalar potentials based on sl(2)Lie algebra is studied.According to the quasi-exact solvability theory,we construct the configuration of the classes II,IV,V,and X potentials in the Turbiner’s classification such that the Dirac equation with scalar potential is quasi-exactly solved and the Bethe ansatz equations are derived in order to obtain the energy eigenvalues and eigenfunctions.     

Quaternion Dirac Equation and Supersymmetry

Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.     

Dirac Equation and Some Quasi-Exact Solvable Potentials in the Turbiner’s Classification

In this paper quasi-exact solvability (QES) of Dirac equation with some scalar potentials based on sl(2) Lie algebra is studied. According to the quasi-exact solvability theory, we construct the configuration of the classes II, IV, V, and X potentials in the Turbiner's classification such that the Dirac equation with scalar potential is quasi-exactly solved and the Bethe ansatz equations are derived in order to obtain the energy eigenvalues and eigenfunctions.     

Exactly Solvable Combinations of Scalar and Vector Potentials for the Dirac Equation Interrelated by Riccati Equations

New classes of solvable scalar and vector potentials for the Dirac equation are obtained, together with the associated exact Dirac spinors. The method of derivation is based on an a priori constraint between the solutions, leading to an interrelation between the scalar and vector potential in the form ofa Riccati equation. The present note generalizes a series of former articles.     

Energy-Levels Crossing and Radial Dirac Equation: Supersymmetry and Quasi-Parity Spectral Signatures

The (3+1)-dimensional Dirac equation with position dependent mass in 4-vector electromagnetic fields is considered. Using two over-simplified examples (the Dirac-Coulomb and Dirac-oscillator fields), we report energy-levels crossing as a spectral property or as an effect of the hidden supersymmetric quantum mechanical language and/or quasi-parity signatures. Under different settings of the related interactions’ way-of-coupling into Dirac equation, it is observed that the two ultimate/effective descendents, Dirac-Coulomb and Dirac-oscillator, exhibit different conditions on the energy-levels crossings.     

Effective-Mass Schrödinger Equation and Generation of Solvable Potentials

A one-dimensional Schrödinger 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 Pöschl-Teller potentials are obtained. Consistency with the intertwining condition is pointed out.     

Supersymmetry and SWKB Approach to Dirac Equation with Hyperbolic Scarf Potential

By using the supersymmetric quantum mechanics and shape invariance concept, we study the Dirac equation with the hyperbolic Scarf potential and the exact energy spectrum is obtained. Also, we calculate the bound state energy eigenvalues by using the supersymmetric WKB approximation approach so that we get the same results.     

具有Poschl-Teller型标量势与矢量势的Klein-Gordon方程和Dirac方程的束缚态

给出了具有Poschl-Teller型标量势与矢量势的Klein-Gordon方程和Dirac方程的s波束缚态解.     

D-Dimensional Dirac Equation for Energy-Dependent Pseudoharmonic and Mie-type Potentials via SUSYQM

We investigate the approximate solution of the Dirac equation for energy-dependent pseudoharmonic and Mie-type potentials under the pseudospin and spin symmetries using the supersymmetry quantum mechanics. We obtain the bound-state energy equation in an analytical manner and comment on the system behavior via various figures and tables.     

Kramers Equation and Supersymmetry

Hamilton’s equations with noise and friction possess a hidden supersymmetry, valid for time-independent as well as periodically time-dependent systems. It is used to derive topological properties of critical points and periodic trajectories in an elementary way. From a more practical point of view, the formalism provides new tools to study the reaction paths in systems with separated time scales. A ‘reduced current’ which contains the relevant part of the phase space probability current is introduced, together with strategies for its computation.     

Exact Solutions of Effective Mass Dirac Equation with Non-PT-Symmetric and Non-Hermitian Exponential-type Potentials

By using a two-component approach to the one-dimensional effective mass Dirac equation, bound states are investigated under the effect of two new non-PT-symmetric and non-Hermitian exponential type potentials. It is observed that the Dirac equation can be mapped into a Schrodinger-like equation by rescaling one of the two Dirac wave functions in the case of the position-dependent mass. The energy levels and the corresponding Dirae eigenfunctions are found analytically.     

Ground State and Spin-Glass Phase of the Large-N Infinite-Range Spin Glass via Supersymmetry

The large-N infinite-range spin glass is considered, in particular, the number of spin components k needed to form the ground state and the sample-to-sample fluctuations in the Lagrange multiplier field on each site. The physical significance of k for the correlation functions is discussed. The difference between the large-N and spherical spin glass is emphasized; a slight difference between the average Lagrange multiplier of the large-N and spherical spin glasses is derived, leading to a slight increase in the energy of the ground state compared to the naive expectation. Further, there is a change in the low-energy density of excitations in the large-N system. A form of level repulsion, similar to that found in random matrix theory, is found to exist in this system, surviving interactions. Even though the system is an interacting one, a supersymmetric formalism is developed to deal with the problem of averaging over disorder.     

Dynamical Symmetries of Two-Dimensional Dirac Equation with Screened Coulomb and Isotropic Harmonic Oscillator Potentials

In this paper, we study symmetrical properties of two-dimensional (2D) screened Dirac Hydrogen atom and isotropic harmonic oscillator with scalar and vector potentials of equal magnitude (SVPEM). We find that it is possible for both cases to preserve so(3) and su(2) dynamical symmetries provided certain conditions are satisfied. Interestingly, the conditions for preserving these dynamical symmetries are exactly the same as non-relativistic screened Hydrogen atom and screened isotropic oscillator preserving their dynamical symmetries. Some intuitive explanations are proposed.     

The (1+1) Dimensional Dirac Equation With Pseudoscalar Potentials: Path Integral Treatment

The supersymetric path integrals in solving the problem of relativistic spinning particle interacting with pseudoscalar potentials is examined. The relative propagator is presented by means of path integral, where the spin degrees of freedom are described by odd Grassmannian variables and the gauge invariant part of the effective action has a form similar to the standard pseudoclassical action given by Berezin and Marinov. After integrating over fermionic variables (Grassmannian variables), the problem is reduced to a nonrelativistic one with an effective supersymetric potential. Some explicit examples are considered, where we have extracted the energy spectrum of the electron and the wave functions. PACS numbers: 03.65. Ca-Formalism, 03.65. Db-Functional analytical methods, 03.65. Pm-Relativistic wave equations.     

非简谐振子与新条件精确可解势

利用超对称性,由平移的非简谐振子构造出了多类新的条件精确可解势,并得到相应的非线性谱生成代数.     

Complex Geometry and Dirac Equation

Complex geometry represents a fundamentalingredient in the formulation of the Dirac equation bythe Clifford algebra. The choice of appropriate complexgeometries is strictly related to the geometricinterpretation of the complex imaginary unit . We discuss two possibilities which appearin the multivector algebra approach: the₁₂₃ and ₂₁ complexgeometries. Our formalism provides a set of rules which allows an immediate translation between thecomplex standard Dirac theory and its version withingeometric algebra. The problem concerning a doublegeometric interpretation for the complex imaginary unit is also discussed.     

On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions

The paper considers macroscopic behavior of a Fermi–Dirac particle system. We prove the L ¹-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box with the collision kernel b(cos θ)|ρ−ρ _*|^γ, which corresponds to very soft potentials: −5 < γ ≤ −3 with a weak angular cutoff: ∫₀ ^π b(cos θ)sin ³θ dθ < ∞. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff–Young inequality, the L ^∞-bounds of the solutions, and a specific property of the value-range of the exponent γ. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy.     

An Alternative Method for Calculating Bound-State of Energy Eigenvalues of Klein-Gordon for Quasi-exactly Solvable Potentials

We obtain the bound-state energy of the Klein-Gordon equation for some examples of quasi-exactly solvable potentials within the framework of asymptotic iteration method （AIM）. The eigenvalues are calculated for type- 1 solutions. The whole quasi-exactly solvable potentials are generated from the defined relation between the vector and scalar potentials.     

Perturbed Coulombic Potentials in Dirac and Klein-Gordon Equations

A relativistic extension of our pseudo-perturbative shifted l-expansion technique is presented to solve for the eigenvalues of Dirac and Klein-Gordon equations. Once more we show the numerical usefulness of its results via comparison with available numerical integration data.