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Journal of Physics A Mathematical and General

Perturbed harmonic oscillator ladder operators: eigenenergies and eigenfunctions for the X2+ λX2/(1+gX2) interaction

N Bessis, Georges Bessis, Gisèle Hadinger

Published Feb 21, 1983• 22 citations
Publisher

Abstract

The perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of Hermite polynomials H k , when V(X)=b 2 X 2 + Sigma k C k H k (b 1 2/X). It is found that the use of a Hermite polynomials basis, together with the use of binomial coefficient functions in the quantum number, greatly simplifies the determination of the perturbed ladder and factorisation functions. Thus, one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation, without increasing intricacy. Thorough calculation has been given for a perturbing potential V(X) function even in X. As an illustrative application of the procedure, the resolution of the Schrodinger equation with a potential function V(X)=X 2 + lambda X 2 /(1+gX 2 ), g>0 is reinvestigated.

DOI
10.1088/0305-4470/16/3/010

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The perturbed ladder operators method is applied to the resolution of the perturbed harmonic oscillator wave equation for the case where the perturbation is expandable in a convergent series of Hermite polynomials H k , when V(X)=b 2 X 2 + Sigma k C k H k (b 1 2/X). It is found that the use of a Hermite polynomials basis, together with the use of binomial coefficient functions in the quantum number, greatly simplifies the determination of the perturbed ladder and factorisation functions. Thus, one obtains analytical expressions of the eigenenergies and eigenfunctions up to any order of the perturbation, without increasing intricacy. Thorough calculation has been given for a perturbing potential V(X) function even in X. As an illustrative application of the procedure, the resolution of the Schrodinger equation with a potential function V(X)=X 2 + lambda X 2 /(1+gX 2 ), g>0 is reinvestigated.
The interaction potential with the form as V(x)= x2+λx2/(1+gx2) where g >0, appears in several areas of laser theory, quantum field theory, atom and nuclear physics. One could consider that the solution of the eigenequation either by the classical Rayleigh-Schr?dinger perturbation scheme or by the perturbed ladder operators scheme. Nevertheless, the perturbation series does not converge for any values of λ and g. In the present paper, it is shown that this difficulty can be overcome as long as the potential function can be expanded in a convergent series on the basis ofthe Hermite polynomials. Therefore, the eigenequation ((d2)/(dx2)-V(x)+ξ)φ(x)=0,∞2)/(dX2)-b2X2-Σkc2kH2k(b1/2X)+ξ)φ(X)=0.
By mapping the perturbation scheme onto the ladder operator formalism, the field of application of the Schrodinger-Infeld-Hull factorisation method is enlarged. It is shown how, at each order of the perturbation, perturbed ladder operators can be constructed. Thus, without having to calculate explicitly either the excited unperturbed functions or any matrix element, one obtains analytical expressions of the perturbed eigenvalues in terms of the quantum numbers of the factorisable unperturbed problem. A three-terms recurrence relation, valid at any rank of the perturbation, is derived and leads to closed form expressions of the perturbed eigenfunctions. Consequently, a closed form expression of any matrix element on the basis of the perturbed eigenfunctions is easily obtained from the calculation of one unique particular integral.
This study presents a complete algebraic formulation of the quantum harmonic oscillator, demonstrating that the Hermite polynomials, H n ( x ) , emerge as the fundamental solution when the power-series method is applied to the Schrödinger equation. The physical requirement of square integrability enforces the truncation of the series, leading precisely to these polynomials, which constitute the polynomial part of the energy eigenfunctions, ψ n ( x )= N n H n ( αx ) e − α 2 x 2 /2 , where n is the quantum number. This quantum number directly determines the quantized energy levels through E n =ℏω( n+ 1 2 ) . The analysis based on creation and annihilation operators further reveals that the action of the creation operator on the ground state generates the excited states along with their corresponding Hermite functions. The orthogonality of these polynomials ensures the orthonormality of the complete set of eigenfunctions. Additionally, the geometric Berry phase emerges as a natural extension when the system undergoes adiabatic cyclic changes, revealing a deeper topological structure that complements the conventional algebraic and analytic descriptions. Therefore, Hermite polynomials encapsulate both the mathematical structure and the physical essence of quantization in this fundamental system.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
The eigenfunctions and eigenvalues of the three-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square interaction are obtained. A realization of the ladder operators for the wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρd/dρ with ρ = r 2 are evaluated. Another hidden symmetry explores the relations between the eigenvalues and eigenfunctions for substituting r→ ir. PACS number(s): 03. 65. Fd, 03. 65.Ge and 02. 20.Qs.