Studies on the Bound-State Spectrum of Hyperbolic Potential

Roy, Amlan K.

doi:10.1007/s00601-013-0767-1

Cited by 9 publications

(8 citation statements)

References 28 publications

(56 reference statements)

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“…In above derivation, we have adopted a proper approximation to the nonlinear centrifugal term

\frac{1}{r^{2}} \approx \frac{α^{2}}{{sinh}^{2} α r} = \frac{α^{2} (1 - y)}{y},

similar to our previous work in studying the spin symmetry problem with this potential and others . In fact, such an approximation first proposed by Greene and Aldrich was used to generate pseudo‐Hulthén wave function for an arbitrary l ‐state.…”

Section: Algebraic Methods To Energy Levelsmentioning

confidence: 99%

Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Wei

Wen-Li

2014

Int J of Quantum Chemistry

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Within the framework of supersymmetric quantum mechanics method, we study by an algebraic method the arbitrary l-wave bound states of the Schr€ odinger equation for the hyperbolical molecular potential by a proper approximation to nonlinear centrifugal term. The explicitly analytical formula of energy levels is derived, and the corresponding bound state wave functions are presented. The function analysis method is used to rederive the same energy levels of the quantum system under consideration to check the validity of this algebraic method. In addition, it is shown from numerical results of energy levels that above certain a parameter depending on the choices of potential parameters V 1 and V 2 the hyperbolical molecular potential cannot trap a particle as it becomes weaker and the energy level starts to turn positive when the potential parameter a becomes larger.

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“…In above derivation, we have adopted a proper approximation to the nonlinear centrifugal term

\frac{1}{r^{2}} \approx \frac{α^{2}}{{sinh}^{2} α r} = \frac{α^{2} (1 - y)}{y},

Section: Algebraic Methods To Energy Levelsmentioning

confidence: 99%

Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Wei

Wen-Li

2014

Int J of Quantum Chemistry

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“…In this work, the GPS method has been used to calculate eigenvalues and eigenfunctions of these states. It has provided highly accurate results for various model and real systems including atoms, molecules, some of which could be found in the references [61,62,63,64,65,66,46,67,68,69,70,71,72,73,74,75,76,47,77,78].…”

Section: Dchomentioning

confidence: 99%

Information analysis in free and confined harmonic oscillator

Mukherjee¹,

Roy²

2021

Preprint

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In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator (CHO). Confined quantum systems have provided appreciable interest in areas of physics, chemistry, biology, etc., since its inception. A particle under extreme pressure environment unfolds many fascinating, notable physical and chemical changes. The desired effect is achieved by reducing the spatial boundary from infinity to a finite region. Similarly, in the last decade, information measures were investigated extensively in diverse quantum problems, in both free and constrained situations. The most prominent amongst these are: Fisher information (I), Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T ), Onicescu energy (E) and several complexities. Arguably, these are the most effective measures of uncertainty, as they do not make any reference to some specific points of a respective Hilbert space. These have been invoked to explain several physico-chemical properties of a system under investigation. Kullback-Leibler divergence or relative entropy describes how a given probability distribution shifts from a reference distribution function. This characterizes a measure of discrimination between two states. In other words, it extracts the change of information in going from one state to another.The one-dimensional confined harmonic oscillator (1DCHO), defined by v), can be classified into two forms, (a) symmetrically confined harmonic oscillator (SCHO) (when d m = 0), (b) asymmetrically confined harmonic oscillator (ACHO) (corresponding to d m = 0). Further, in latter case, confinement can be accomplished two different ways: (i) by changing the box boundary, keeping box length and d m fixed at zero; (ii) another route is to adjust d m by keeping box length and boundary fixed. SCHO can be treated as an intermediate between a particle-ina-box (PIB) and a 1DQHO. Though the Schrödinger equation for SCHO can be solved exactly, for ACHO it cannot be. We have employed two different methods for the latter, viz., (i) an imaginary time propagation (ITP), leading to minimization of an expectation value (ii) a variation-induced exact diagonalization procedure that utilizes SCHO eigenfunctions as basis. It is found that, at very low x c region, I, S, R, T, E remain invariant with change confinement length, x c . At moderate x c region S x , R x , T x progress and I x , E x decrease with rise in x c . Additionally, special attention has been paid to study relative information in SCHO and ACHO.Analogously, a 3DCHO (radically confined within an impenetrableFree and confined harmonic oscillator • • • spherical well) can act as a bridge between particle in a spherical box (PISB) and isotropic 3DQHO. The time-independent Schrödinger equation for D-dimensional harmonic oscillator in both free and confined condition can be solved exactly, within a Dirichlet boundary condition. Here a detailed exploration of information measures has been carried out in r and p spaces, for a 3DCHO. Some ex...

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“…There are a few method to solve the second‐order differential equation: function analysis method, exact quantization rule method, supersymmetric quantum mechanics method, factorization method, asymptotic iteration method, generalized pseudospectral method, pseudospectral method, variational method, perturbation method, shifted 1∕ N expansion method . These methods have some advantages and/or disadvantages to find exact eigenvalues equation and eigenfunctions according to each other.…”

Section: Schrödinger Equation For Htm Potential Within the New Develomentioning

confidence: 99%

Bound and scattering states for a hyperbolic‐type potential in view of a new developed approximation

Aydoğdu

Yanar

2015

Int J of Quantum Chemistry

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A new developed approximation is used to obtain the arbitrary l-wave bound and scattering state solutions of Schr€ odinger equation for a particle in a hyperbolic-type potential. For bound state, the energy eigenvalue equation and unnormalized wave functions in terms of Jacobi polynomials are achieved using the Nikiforov-Uvarov (NU) method. Besides, energy eigenvalues are calculated numerically for some states and compared with those given in the literature to check accuracy of our results.For scattering state, the wave function is found in terms of hypergeometric functions. Furthermore, scattering amplitude and phase shifts are achieved using scattering solutions. Also it is shown that the energy eigenvalue equation obtained from analytic property of scattering amplitude is same with one obtained using NU method.

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Studies on the Bound-State Spectrum of Hyperbolic Potential

Cited by 9 publications

References 28 publications

Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Information analysis in free and confined harmonic oscillator

Bound and scattering states for a hyperbolic‐type potential in view of a new developed approximation

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