Exact solutions of the rigid rotor in the electric field

Abstract

We first present a new constraint condition on the confluent Heun function HC(α, β, γ, δ, η;z) (β, γ ≥ 0, z ∈ [0, 1]) and then illustrate how to solve the rigid rotor in the electric field. We find its exact solutions unsolved previously through solving the Wronskian determinant. The present results compared with those by the perturbation methods are found to have a big difference for a large parameter a. We also present 2D and 3D probability density distributions by choosing different angular momentum quantum numbers l. We observe that the original eigenvalues with degeneracy (2 l + 1) are split into the (l + 1) state with approximate eigenvalues l(l + 1) for small a but large l.

Exact solutions of the Schrödinger equation for a class of hyperbolic potential well

“…where n is the number of the nodes of the wave function. This paper is organized as follows: In Section 2 based on our recent works, [29][30][31][32][33] taking U 1 as a typical example, we describe how to accurately solve the bound states of the Schrödinger equation. For this purpose, the first step is to transform the corresponding Schrödinger equation into a confluent Heun differential equation by using different forms of variable substitution and function transformation and then obtain two linearly dependent solutions for the same eigenstate.…”

Application of Eckart-Hellmann potential to study selected diatomic molecules using Nikiforov-Uvarov-Functional analysis method

“…The development of these methods allows one to derive the analytic eigen-solutions of the relativistic and non-relativistic wave equations which play a crucial role in interpreting the behavior of quantum mechanical systems. The frequently used analytical methods are the Nikiforov-Uvarov method (NU) , Asymptotic iterative method (AIM) [31], Laplace transformation approach [32], ansatz solution method [33], super-symmetric quantum mechanics approach (SUSYQM) [34,35], exact and proper quantization methods [36,37], the series expansion method [38][39][40][41][42][43][44][45], and the recent study via the Heun function approach has been used widely to study those soluble quantum systems which could not be solved before, such as the systems including the Mathieu potential, rigid rotor problem, sextictype problem, or the Konwent potential, to name a few [46][47][48][49][50][51][52][53][54].…”

“…The frequently used analytical methods are the Nikiforov-Uvarov method (NU) , Asymptotic iterative method (AIM) [31], Laplace transformation approach [32], ansatz solution method [33], super-symmetric quantum mechanics approach (SUSYQM) [34,35], exact and proper quantization methods [36,37], series expansion method [38][39][40][41][42][43][44][45], the recent study via the Heun function approach has been used widely to study those soluble quantum systems which could not be solved before,e.g. the systems including the Mathieu potential,rigid rotor problem,sextic type problem, Konwent potential and others [46][47][48][49][50][51][52][53][54] The Schrödinger equation (SE) can be studied for different quantum-mechanical processes with the above analytical methods [55][56][57][58]. The analytical solutions to this equation with a physical potential plays an important role in our understanding of the fundamental root of a quantum system.…”

Schrödinger equation based analytic assessment of thermal properties of confined electrons under vector and scalar fields

“…[37][38][39] However, a large number of potentials of physical importance used into the Schrödinger equation or the perturbations into Dirac and Dirac-Weyl equations may be transformed into the form of the biconfluent Heun function (BHF). [40][41][42][43][44][45][46][47][48] Moreover, from a century ago, the Schrödinger equation can be reduced to the BHF known in mathematics for harmonium. 49,50 Although, some of the authors have submitted different solvable traditional models as analytically, [51][52][53][54][55] and there are a few works about of the geometric model in these areas.…”

“…It is interesting that nearly all analytic solutions of the nonrelativistic equation have been expressed in terms of hypergeometric functions 37–39 . However, a large number of potentials of physical importance used into the Schrödinger equation or the perturbations into Dirac and Dirac–Weyl equations may be transformed into the form of the biconfluent Heun function (BHF) 40–48 …”