Dimensional Schrodinger Equation Research Articles

The pseudoharmonic oscillator energy spectrum

The solution to the one dimensional Schrodinger equation for the potential of type x 2 + αx −2, for positive values of the constant α is discussed using an operator method by factorizing the Hamiltonian.

Birkhoff normal forms for Hamiltonian PDEs in their energy space

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that, to exchange energy, modes have to oscillate at the same frequency. Contrary to the previous existing results, we do not require the solutions to be especially smooth. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d = 1 and nonlinear Schrodinger equations in dimension d ≤ 2.

A variety of new periodic solutions to the damped [formula omitted]-dimensional Schrodinger equation via the novel modified rational sine–cosine functions and the extended tanh–coth expansion methods

In this work, new interesting types of periodic solutions to the damped (2+1)-dimensional Schrodinger equation are extracted by applying new modifications of both rational sine–cosine and sinh–cosh techniques, and also by implementing the extended tanh–coth expansion method. Furthermore, the significance of the embedded nonlinearity parameter and the damping factor acting on the propagations of the obtained solutions to the proposed equation is explored.

Wave Structures for Nonlinear Schrodinger Types Fractional Partial Differential Equations Arise in Physical Sciences

Nonlinear partial differential equations are mostly renowned for depicting the underlying behavior of nonlinear phenomena relating to the nature of the real world. In this paper, we discuss analytic solutions of fractional-order nonlinear Schrodinger types equations such as the space-time fractional nonlinear Schrodinger equation and the (2+1)-dimensional time-fractional Schrodinger equation. The considered equations are converted into ordinary differential equations with the help of wave variable transformation and then the recently established rational ( )-expansion method is employed to construct the exact solutions. The obtained solutions have appeared in the forms of a trigonometric function, hyperbolic function, and rational function which are compared with those of literature and claimed to be different. The graphical representations of the solutions are finally brought out for their physical appearances. The applied method is seemed to be efficient, concise, and productive which might be used for further research.
 Mathematics Subject Classifications: 35C08, 35R11

A new (3 + 1)-dimensional Schrödinger equation: derivation, soliton solutions and conservation laws

In the present paper, a new (3 + 1)-dimensional Schrodinger equation in Quantum Mechanics is derived. Based on the extended (3 + 1)-dimensional zero curvature equation, this equation is derived for the first time via the compatibility condition. Meanwhile, some soliton solutions are presented. Finally, conservation laws also obtained.

Solution of the Ultra Generalized Exponential–Hyperbolic Potential in Multi-dimensional Space

In this work, we proposed ultra generalized exponential–hyperbolic potential (UGEHP) and derived various well known exponential–hyperbolic type potentials by setting parameters in UGEHP and using approximation suggested by Greene–Aldrich. The bound state solutions of the multi (D)-dimensional Schrodinger equation for UGEHP have been presented using the parametric Nikiforov–Uvarov method. The approximate analytical bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. We also investigated the rotational vibrational (RV) partition function from the eigenvalue of UGEHP. By the setting parameters, we obtained eigenvalue spectrum and RV partition function for the screened cosine Kratzer potential, screened Kratzer potential, attractive radial potential, quadratic exponential-type potential, Manning Rosen with class of Yukawa potential and Yukawa potential, class of Yukawa potential, mixed class of Yukawa potential, quantum interaction potential or Hulthen–Yukawa inversely quadratic potential, Hulthen plus inversely quadratic exponential Mie-type potential and Hulthen plus exponential Coulombic potential with centrifugal potential barrier. We studied the behavior of energy eigenvalues and RV partition function for the UGEHP. We computed and tabulated numerical results for $$CO, NO, I_2$$ , HCl and LiH diatomic molecules and compared with numerical results available in literature for same diatomic molecules.

Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion

<p style='text-indent:20px;'>In this article, the asymptotic behavior of the solution to the following one dimensional Schrödinger equations with white noise dispersion <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ idu + u_{xx}\circ dW+ |u|^{p-1}udt = 0 $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>is studied. Here the equation is written in the { Stratonovich} formulation, and <inline-formula><tex-math id="M1">\begin{document}$ W(t) $\end{document}</tex-math></inline-formula> is a standard real valued Brownian motion. After establishing the global well-posedness, theoretical proof and numerical investigations are provided showing that, for a deterministic small enough initial data in <inline-formula><tex-math id="M2">\begin{document}$ L^1_x\cap H^1_x $\end{document}</tex-math></inline-formula>, the expectation of the <inline-formula><tex-math id="M3">\begin{document}$ L^\infty_x $\end{document}</tex-math></inline-formula> norm of the solutions decay to zero at <inline-formula><tex-math id="M4">\begin{document}$ O(t^{-\frac14}) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>, as soon as <inline-formula><tex-math id="M7">\begin{document}$ p&gt;7 $\end{document}</tex-math></inline-formula>.

Lie Algebras of Heat Operators in a Nonholonomic Frame

We construct the Lie algebras of systems of $$2g$$ graded heat operators $$Q_0,Q_2,\dots,Q_{4g-2}$$ that determine the sigma functions $$\sigma(z,\lambda)$$ of hyperelliptic curves of genera $$g=1$$ , $$2$$ , and $$3$$ . As a corollary, we find that the system of three operators $$Q_0$$ , $$Q_2$$ , and $$Q_4$$ is already sufficient for determining the sigma functions. The operator $$Q_0$$ is the Euler operator, and each of the operators $$Q_{2k}$$ , $$k>0$$ , determines a $$g$$ -dimensional Schrodinger equation with potential quadratic in $$z$$ for a nonholonomic frame of vector fields in the space $$\mathbb C^{2g}$$ with coordinates $$\lambda$$ . For any solution $$\varphi(z,\lambda)$$ of the system of heat equations, we introduce the graded ring $$\mathscr R_\varphi$$ generated by the logarithmic derivatives of $$\varphi(z,\lambda)$$ of order $$\ge 2$$ and present the Lie algebra of derivations of $$\mathscr R_\varphi$$ explicitly. We show how this Lie algebra is related to our system of nonlinear equations. For $$\varphi(z,\lambda)=\sigma(z,\lambda)$$ , this leads to a well-known result on how to construct the Lie algebra of differentiations of hyperelliptic functions of genus $$g=1,2,3$$ .

Solution of d-dimension time independent cosmic string using supersymmetry quantum mechanics method for rosen morse, scarf II and scarf I non-central potentials

Solution of d-dimension time independent cosmic string for rosen morse, scarf II and scarf I non-central potentials was solved using supersymmetry quantum mechanics method. Variable separation method was applied to the d-dimension time independent cosmic string equation, and then the d-dimension time independent cosmic string equation was reduced into d one dimensional schrodinger equation. Supersymmetry quantum mechanics method was used to analyze radial wave function, angular wave functions, and energy level. Energy levels were higher by the presence of rosen morse, scarf II and scarf I potentials parameters. The increased value of nr , n θ 2 and n θ 4 caused the increase in energy levels, the decrease of cosmic string α parameter, caused the decrease of energy levels and the increasing of number n θ 1 and n θ 3 caused the decrease of energy levels.

Construction of Functional Closed Form Wave Solutions to the ZKBBM Equation and the Schrödinger Equation

The time fractional (1+1)-dimensional Schrodinger equation and the space–time fractional (1+1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation are important mathematical models to interpret signal processing in optical fibers, the fractional quantum mechanics, water waves due to gravity, the motion of turbulent flows, the wave of driving flow of fluid, the ion osculate waves in plasma, etc. In the present article, we have examined the closed form soliton solutions with the assistance of the modified simple equation method together with the fractional wave transformation. The form of the attained solutions is rational, trigonometry and hyperbolic functions. We have shown that the assigned method is further general, efficient, straightforward and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering. We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.

The new solitary wave structures for the (2 + 1)-dimensional time-fractional Schrodinger equation and the space-time nonlinear conformable fractional Bogoyavlenskii equations

The present paper employs the space-time fractional nonlinear Bogoyavlenskii equation and Schrodinger equation. We perform a new method to take some new solitary wave phenomena for each equation. Those solutions can explain through hyperbolic, trigonometric and rational functions. The graphical design makes the dynamics of the equations noticeable. Herein, the intended approach is simplistic, conventional, and convenient in implementing many solitary wave phenomena of several nonlinear fractional wave equations occurring in mathematical physics and engineering as well.

Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order <i>α</i> and <i>β</i>. Moreover, numerical solutions are demonstrated to justify the theoretical results.

Several exact solutions for three dimensional Schrodinger equation involving inverse square and power law potentials

Several exact solutions for three dimensional Schrodinger equation involving inverse square and power law potentials

Solving Schrödinger Equation for Finite Potential Well Using the Iterative Method

The quantum Finite Square Well (FSW) model is a well-known topic in most quantum mechanics (QM) books. A couple of equations can be derived from one dimensional Schrodinger equation for a finite potential well for describing the bound Eigen states within the well. Sometimes the FSW problem does not have an exact solution, yet, there are in fact exact solutions. In this work a computational techniques is adopted to find the exact solution for FSW. To achieve this computational solution, a computer program has been written in Basic language for calculating the Eigen state energy for a particle confined in finite potential well using the iterative method (IM). Six values of potential width have been studied with potential depth. The results showed that wider potential width led us to more bound states, while narrower potential width led us to less bound states. Six values of potential depth have been studied with potential width. The results showed that larger potential depth led us to more bound states, while smaller potential width led us to less bound states. A Comparison Between Finite and Infinite Potential Well has been also presented. The result of comparison showed that energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. In general, the matching between the results of the iterative method and the graphical method (GM) proving that the iteration method can be regarded as a useful tool for describing the solutions of the 1-dimensional FSW problem.

Solution of time independent cosmic string for harmonic oscillator plus poschl teller non-central potentials using supersymmetry quantum mechanics methods

Time independent cosmic string equation for Harmonic Oscillator and Poschl Teller non-central potential have been solved using supersymmetry quantum mechanics method. The time independent cosmic string with non-central potential was reduced into three one dimensional Schrodinger equations by using variable separation method. The energy levels and radial wave function were analyzed using supersymmetry quantum mechanics method from radial part of Schrodinger like equation. The angular wave function was obtained from angular part of Schrodinger equation. The effect of the non-central potential caused energy levels increasing. Energy spectra for λ = 0 and κ = 0 have smaller value than energy levels for λ ≠ 0 and κ ≠ 0. The increased value of parameter λ and κ caused increased in energy levels. However, the decreased of cosmic string parameter α, caused the increased of energy levels.

Traveling wave solutions for complex nonlinear space–time fractional order (2 + 1)-dimensional Maccari dynamical system and Schrödinger equation with dual power law nonlinearity

The traveling wave solutions for complex (2 + 1)-dimensional Maccari dynamical system and Schrodinger equation in both space and time fractional terms are studied in this paper. Using extended simple equation method, exact, periodic and soliton solutions in fractional form are extracted for both models under study confirming the effectiveness of the method. The obtained fractional results have a variety of applications in fibre optics, quantum mechanics, oceanic and optical sciences.

On the motion of non-relativistic particle in doubled ring-shaped Coulomb oscillator potential within cosmic string framework

The spacetime metric of cosmic string has been used to get the new form of Laplacian in Schrodinger equation for doubled ring-shaped Coulomb oscillator (DRSCO) potential. We use separation variable method to get three of one dimensional Schrodinger equations. Radial form is the main idea of this problem, so we just point out the solution of one dimensional Schrodinger equation for the radial part. Supersymmetric quantum mechanic method is used to get the energy spectrum and the wave function and we have used the approximation in the calculation of superpartner potential to get the same variable function as sextic-like potential in the radial form.

Exact Solutions of Schrodinger Equation in Cylindrical Coordinates for Double Ring-Shaped Coulomb Oscillator Potential Using SUSY QM Method

Schrodinger equation for a doubled ring-shaped coulomb oscillator potential is investigated using supersymmetric quantum mechanics approach. The three dimensional Schrodinger equation for a doubled ring-shaped coulomb oscillator potential in cylindrical coordinates is separated into three parts that contains one dimensional Schrodinger type equation which are solved using supersymmetric operator and the idea of shape invariant potential. The energy spectrum and the total wave functions are obtained.

A meshless method of line radial base function study of Gaussian wave packet broadening in few semiconducting mediums: electron–electron interaction effects

In this paper, we have studied the effect of electron–electron interaction on wave packet broadening in different semiconducting mediums in the presence of conduction band non-parabolicity. We have solved the resulting one dimensional fourth order Schrodinger equation by means of a meshless radial base function approach and 2nd order Runge Kutta method. We have compared different semiconducting mediums GaAs, GaN, AlN, InSb and GaSb and showed that in the absence of the electron–electron interaction, the Gaussian wave packet decays with time elapse while in the presence of the electron–electron interaction, the Gaussian wave packet localizes when time increases. Finally, Gaussian wave packet also localizes faster when we increase the electron–electron interaction strength.

A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures

In this paper we derive by an entropy minimization technique a local Quantum Drift-Diffusion (QDD) model that allows to describe with accuracy the transport of electrons in confined nanostructures. The starting point is an effective mass model, obtained by considering the crystal lattice as periodic only in the one dimensional longitudinal direction and keeping an atomistic description of the entire two dimensional cross-section. It consists of a sequence of one dimensional device dependent Schrodinger equations, one for each energy band, in which quantities retaining the effects of the confinement and of the transversal crystal structure are inserted. These quantities are incorporated into the definition of the entropy and consequently the QDD model that we obtain has a peculiar quantum correction that includes the contributions of the different energy bands. Next, in order to simulate the electron transport in a gate-all-around Carbon Nanotube Field Effect Transistor, we propose a spatial hybrid strategy coupling the QDD model in the Source/Drain regions and the Schrodinger equations in the channel. Self-consistent computations are performed coupling the hybrid transport equations with the resolution of a Poisson equation in the whole three dimensional domain.