Exact and approximate solutions of Schrödinger’s equation for a class of trigonometric potentials

Abstract

Abstract:The asymptotic iteration method is used to find exact and approximate solutions of Schrödinger's equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schrödinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.PACS (

Heavy quarkonia properties from a hard-wall confinement potential model with conformal symmetry perturbing effects

“…Introducing a suitable transformation as u nℓ (r/a) = sin(r/a)f (r/a), which ensures that u nℓ (0) = u nℓ (π) = 0, followed by a change of variable, y = cot r a , and upon making use of the Ansatz [12] f (y) = (1 + y 2 )…”

“…Before proceeding further, we like to notice that in being hard walled, the potential under discussion, has frequently found applications to the studies of systems featuring confinement and ranging from quantum dots [8], over Coulomb fluids [9], to hadron structure [7], [10]. Furthermore, in [12], the same potential has been considered as belonging to a broader class of trigonometric interactions, to which exact and approximate solutions have been found by means of the asymptotic iteration method. In addition, in [13], the trigonometric Rosen-Morse potential and its exact solutions have served as a point of departure towards the constructions of new potentials by subjecting it to transformations for which bound states spectra and the corresponding wave functions have been calculated.…”

Evaluated Excited-State Time-Independent Correlation Function and Eigenfunction of the Harmonics Oscillator Cosine Asymmetric Potential via Numerical Shooting Method

“…To solve this partial differential equation, we need to specify the potential V( ) as well as the boundary condition; the boundary condition can be obtained from the physical requirement of the system. Suppose a particle is bound state to around of attraction by the harmonics oscillator Cosine asymmetric potential (see Figure 1): with + cos ( ) that is called the Cosine asymmetric potential [22], where , , and are positive constants. Substituting the harmonics oscillator Cosine asymmetric potential from (3) into (2) leads to the following equation:…”

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We aimed to evaluate the ground-state and excite-state energy eigenvalue (En), wave function, and the time-independent correlation function of the atomic density fluctuation of a particle under the harmonics oscillator Cosine asymmetric potential (Saad et al 2013). Instead of using the 6-point kernel of 4 Green's function (Cherroret and Skipetrov, 2008), averaged over disorder, we use the numerical shooting method (NSM) to solve the Schrödinger equation of quantum mechanics system with Cosine asymmetric potential.

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Axially symmetrical molecules in electric and magnetic fields: energy spectrum and selection rules

“…These models cover a very wide area of physical problems, such as molecular vibrations, motion of an electron in the field of crystalline lattice in metals or semiconductors, quantum dots confined in parabolic potential, the impact and probable hazards caused by electromagnetic fields on living organisms etc [1][2][3][4][5][6][7][8][9][10]. Various quantum mechanical models for analytical modeling of multi-dimensional atomic and molecular vibrations have been developed and different methods of solving equations have been used [4][5][6][11][12][13][14][15]. In most cases, the actual multi-dimensional problem is observed as a system of one-dimensional problems, in order to reduce the computational effort required to solve the Schrödinger equation in 2 or 3 dimensions.…”