**Electronic states of graphene nanoribbons studied with the Dirac equation**

We look at the electronic states of slender graphene ribbons (“nanoribbons”) with zigzag and armchair edges. The finite width of these systems breaks the spectrum into an countless set of bands, which we exhibit may be quantitatively understood the usage of the Dirac equation with suitable boundary situations. For the zigzag nanoribbon we show that the boundary situation permits a particlelike and a holelike band with evanescent wave features restricted to the surfaces, which continuously change into the well-known zero electricity floor states because the width gets large.** [1]**

**Finite basis sets for the Dirac equation constructed from B splines**

A process is given for constructing foundation units for the radial Dirac equation from B splines. The resulting basis units, which consist of terrible-energy states in a natural way, allow the accurate assessment of the more than one sums over intermediate states occurring in relativistic many-body calculations. Illustrations are given for the Coulomb-field Dirac equation and assessments of the ensuing foundation sets are defined. As an software, relativistic corrections to the second one-order correlation electricity in helium are calculated. Another software is given to decide the spectrum of thallium (wherein finite–nuclear-size outcomes are critical) in a version capacity. Construction of B-spline foundation units for the Dirac-Hartree-Fock equations is described and the resulting basis sets are carried out to examine the cesium spectrum. **[2]**

**Kinks and the Dirac Equation**

In a version quantum theory of interacting mesons, the motion of certain conserved particlelike systems is mentioned. It is proven how collective coordinates may be brought to explain them, main, in lowest approximation, to a Dirac equation. **[3]**

**From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks**

A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through nearby unitaries. In a preceding paper, we confirmed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. We observe a spacetime coordinate transformation upon the lattice of this QW, and show that it’s far equivalent to introducing spacetime-dependent nearby unitaries —at the same time as retaining the lattice constant. By exploiting this duality among adjustments in geometry, and changes in neighborhood unitaries, we display that the spacetime-established QW simulates the Dirac equation in (2 + 1)–dimensional curved spacetime. **[4]**

**Energy Spectrum of the K-State Solutions of the Dirac Equation for Modified Eckart Plus Inverse Square Potential Model in the Presence of Spin and Pseudo-Spin Symmetry within the Framework of Nikifarov-Uvarov Method**

The genuine analytical certain state answers of wave equations are nevertheless very exciting problems in essential quantummechanics. However, there are only a few potentials for which these wave equations can be exactly solved. In this paper, Spin and pseudospin symmetries of the Dirac equation for Modified Eckart plus Inverse square capacity inside a zero tensor interplay are investigated the usage of the parametric Nikiforov-Uvarov method that’s based on the answers of wellknown 2d-order linear differential equations with special capabilities. The sure nation eigen price was received with a few few cases of ability considerations.** [5]**

**Reference**

**[1]** Brey, L. and Fertig, H.A., 2006. Electronic states of graphene nanoribbons studied with the Dirac equation. Physical Review B, 73(23), (Web Link)

**[2]** Johnson, W.R., Blundell, S.A. and Sapirstein, J., 1988. Finite basis sets for the Dirac equation constructed from B splines. Physical Review A, 37(2), (Web Link)

**[3]** Skyrme, T.H.R., 1971. Kinks and the Dirac equation. Journal of Mathematical Physics, 12(8), (Web Link)

**[4]** From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks

Pablo Arrighi, Giuseppe Di Molfetta, Ivan Marquez-Martin & Armando Perez

Scientific Reports volume 9, (Web Link)

**[5]** Louis, H., Ikeuba, A. I., Ita, B. I., Amos, P. I., Magu, T. O., Akakuru, O. U. and Nzeata, N. A. (2018) “Energy Spectrum of the K-State Solutions of the Dirac Equation for Modified Eckart Plus Inverse Square Potential Model in the Presence of Spin and Pseudo-Spin Symmetry within the Framework of Nikifarov-Uvarov Method”, Physical Science International Journal, 17(4), (Web Link)