Solution of the Dirac Equation for the Rectilinear Periodic Motion of an Electron

1969 ◽  
Vol 24 (3) ◽  
pp. 344-349
Author(s):  
A. D. Jannussis
Keyword(s):  
Perturbation Theory ◽  
Dirac Equation ◽  
Periodic Motion ◽  
Periodic Potential ◽  
Bragg Reflection ◽  
Wave Number ◽  
Periodic Functions ◽  
Energy Bands ◽  
Energy Eigenvalues ◽  
The Hill

AbstractIn this paper the Dirac equation for a rectilinear onedimensional periodic potential is treated. It is shown that the energy eigenvalues are periodic functions of the wave number Kϰ and the continuous spectrum is split into energy bands. The end points of the energy bands are the points where the Bragg reflection takes place. These results are obtained by perturbation theory, as well as by the method of determinants, since the resulting eigenvalue equation has the form of a determinant which is similar to the Hill determinant.

Chern Numbers and Green's Functions in Solid State Physics

1997 ◽  
Vol 11 (11) ◽  
pp. 1389-1410
Author(s):  
Xiao-Rong Wu-Morrow ◽  
Cecile Dewitt-Morette ◽  
Lev Rozansky
Keyword(s):  
Periodic Potential ◽  
Solid State Physics ◽  
Hall Conductance ◽  
Energy Eigenvalues ◽  
Finite Dimensional ◽  
Topological Quantum ◽  
Chern Numbers ◽  
Density Response ◽  
Physical Quantities ◽  
Noninteracting Electron

Using the energy Green's function formulation proposed by Niu 1 for particle densities, we construct and clarify the nature of the topological invariant assigned to the Hall conductance in the Hall system of 2-dimensional noninteracting electron gas; we identify this topological quantum number explicitly as the first Chern number of a complex vector bundle over a 2-torus parametrized by the magnetic potential (a1, a2); the fibres are finite dimensional spaces spanned by eigenfunctions of the system with energy eigenvalues below the Fermi energy. Other cases can be treated by a similar procedure, namely, by recognizing that some physical quantities are integrals of curvatures defined on a nontrivial finite dimensional complex bundle. Therefore, in suitable units, they take integer values. We treat, as an example, the electron density response to a dilation of a periodic potential. The integer in this case is the number of Bloch bands. The quantization of the Hall conductance and density response is also shown in the presence of disorder.

Dirac equation with multiparameter-potential and Hulthén tensor interaction under relativistic symmetries

Open Physics ◽  
2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Sami Ortakaya
Keyword(s):  
Dirac Equation ◽  
Spin Symmetry ◽  
Tensor Interaction ◽  
Numerical Results ◽  
Exponential Potential ◽  
Energy Eigenvalues ◽  
Nu Method

AbstractThe pseudospin and spin symmetric solutions of the Dirac equation with Hulthén-type tensor interaction are obtained under multi-parameter-exponential potential (MEP) for arbitrary κ states. The energy eigenvalues and the corresponding eigenfunctions are also obtained using the parametric Nikiforov-Uvarov (NU) method. Some numerical results are also obtained for pseudospin and spin symmetry limits.

scholarly journals Eigen Spectra for q-Deformed Hyperbolic Scarf Potential Including a Coulomb-like Tensor Interaction

2011 ◽  
Vol 3 (2) ◽  
pp. 239-247 ◽  
Author(s):  
M. Eshghi ◽  
H. Mehraban
Keyword(s):  
Dirac Equation ◽  
Spin Symmetry ◽  
Tensor Interaction ◽  
Tensor Coupling ◽  
Energy Eigenvalues ◽  
Scarf Potential ◽  
Tensor Potential ◽  
Uvarov Method

We study the Dirac equation for the q-deformed hyperbolic Scarf potential including a coulomb-like tensor potential under the spin symmetry. The parametric generalization of the Nikiforov-Uvarov method is used to obtain the energy eigenvalues equation and the unnormalized wave functins.Keywords: Dirac equation; q-deformed hyperbolic Scarf; Spin symmetry; Tensor coupling.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i2.7295                 J. Sci. Res. 3 (2), 239-247 (2011)

Borg's Periodicity Theorems for First-Order Self-Adjoint Systems with Complex Potentials

2016 ◽  
Vol 60 (3) ◽  
pp. 615-633 ◽  
Author(s):  
Sonja Currie ◽  
Thomas T. Roth ◽  
Bruce A. Watson
Keyword(s):  
Periodic Potential ◽  
Adjoint System ◽  
Complex Potentials ◽  
Order System ◽  
Compact Sets ◽  
Periodic Functions ◽  
First Order ◽  
Adjoint Systems ◽  
Pauli Matrices

AbstractA self-adjoint first-order system with Hermitian π-periodic potential Q(z), integrable on compact sets, is considered. It is shown that all zeros of are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q(z) is π/2-periodic. Furthermore, the zeros of are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which Q(z) = σ2Q(z)σ2. Here, Δ denotes the discriminant of the system and σ0, σ2 are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if Q = rσ0 + qσ2, for some real-valued π-periodic functions r and q integrable on compact sets.

Dirac Equation with Mixed Scalar–Vector–Pseudoscalar Linear Potential under Relativistic Symmetries

2015 ◽  
Vol 70 (9) ◽  
pp. 713-720 ◽  
Author(s):  
Hadi Tokmehdashi ◽  
Ali Akbar Rajabi ◽  
Majid Hamzavi
Keyword(s):  
Dirac Equation ◽  
Bound States ◽  
Wave Functions ◽  
Linear Potential ◽  
Energy Eigenvalues ◽  
Nu Method

AbstractIn the presence of spin and pseudospin (p-spin) symmetries, the approximate analytical bound states of the Dirac equation, which describes the motion of a spin-1/2 particle in 1+1 dimensions for mixed scalar–vector–pseudoscalar linear potential are investigated. The Nikiforov–Uvarov (NU) method is used to obtain energy eigenvalues and corresponding wave functions in their closed forms.

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