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American Journal of Physics

Electric polarizability and the solution of an inhomogeneous differential equation

M. A. Maize, Chad A. Burkholder

Published Mar 1, 1995• 15 citations
Publisher

Abstract

In this paper, we are going to illustrate the effectiveness of the method of the inhomogeneous differential equation in obtaining the energy shift of a quantum level in second order perturbation theory. The energy shift will be used to calculate the electric polarizability due to the interaction between a static electric field and a charged particle moving under the influence of a one-dimensional delta potential. Both relativistic and nonrelativistic problems will be treated.

DOI
10.1119/1.18092

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In this paper, we are going to illustrate the effectiveness of the method of the inhomogeneous differential equation in obtaining the energy shift of a quantum level in second order perturbation theory. The energy shift will be used to calculate the electric polarizability due to the interaction between a static electric field and a charged particle moving under the influence of a one-dimensional delta potential. Both relativistic and nonrelativistic problems will be treated.
The exactly soluble model, originally considered by Moshinsky, consisting of two spin-1/2 particles interacting through harmonic-oscillator forces and moving in a common harmonic-oscillator potential is extended to include the effects of an external electric field. We solve for the energy exactly, exactly in the Hartree–Fock approximation, and also compare to the results of Hartree–Fock perturbation theory. We find that the Hartree–Fock method yields the same result for the electric polarizability as the exact solution even though the field-independent part of the energy is not the same. However, when second-order Hartree–Fock perturbation theory is employed, the results for the polarizability can be substantially in error. Similar results obtain for the calculation of the matrix element for optical absorption in the dipole approximation.
In this study, the deformed Klein–Gordon equation, Dirac equation, and Schrödinger equations were solved with the improved the modified Mobius square potential model (IMSPM, in short) using Bopp’s shift and standard perturbation theory methods in the symmetries of extended quantum mechanics. By employing the improved approximation to the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some selected diatomic molecules (H2, I2, CO, NO, and HCl). The relativistic shift energy ΔEtotmsp(n,α,A,B,V0,Θ,σ,χ,j,l,s,m) and the perturbative nonrelativistic corrections ΔEnrmsp(n,α,A,B,V0,Θ,σ,χ,j,l,s,m) appeared as a function of the parameters (α,A,B,V0), the parameters of noncommutativity (Θ,σ,χ), in addition to the atomic quantum numbers (n,j,l,s,m). In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result under the improved modified Mobius square potential model is consistent with what is obtained in the literature. In the new symmetries of NCQM, is not possible to get the exact analytical solutions for l=0 and l≠0 can only be solved approximately. Through this research, we came to two noteworthy results, the first is related to the deformed Klein-Gordon equation under the influence of the improved modified Mobius square potential model becomes equivalent to the Duffin-Kemmer equation for a meson with spin-s, while the second result concerns the deformed Schrödinger equation, which can now describe the state of high-energy fermionic particles similar to the Dirac equation in the literature.
Abstract This review presents time-independent perturbative methods for solving the one-dimensional Schrödinger equation, highlighting representative cases that reveal key aspects of the theory. The focus is on their relevance to quantum computing applications, particularly in systems with finite-dimensional state spaces. The main link between these methods and quantum processing emerges when we consider systems with only a finite number of well-defined energy levels, e.g., two levels corresponding to the ‘zero’ and ‘one’ states of a quantum bit (qubit), for which the Schrödinger equation describes a quantum algorithm corresponding to the unitary evolution of the qubits. Perturbative techniques can then be applied to this evolution to gain deeper insight into the algorithm or to simplify it in a controlled manner, as described in the last sections of this review. We initially discuss representative applications using benchmark cases that exemplify the essential features of perturbation theory while avoiding unnecessary technical complications. The methods presented involve direct approaches to solutions for quantum systems with small potential perturbations, with consideration of indirect and approximation techniques. Key cases examined include the particle in a potential well, the harmonic oscillator, and systems with polynomial or Dirac delta distribution perturbations, each chosen to highlight unique aspects of perturbative analysis. Starting with an overview of the unperturbed Schrödinger equation, we derive canonical solutions and then use perturbative techniques to approximate energy shifts and wavefunction corrections in both non-degenerate and degenerate settings. By combining theoretical discussion with case studies, this review underscores how perturbative methods are applied across a range of elementary quantum mechanical problems especially for applications in the qubit model, which, in many cases of interest, can be reduced to the explicit calculations shown here.