Logarithmic Perturbation Theory Research Articles

Semiclassical approach to perturbation theory

The problem of constructing approximate solution methods and finding convenient recurrent algorithms for the study of bound states of quantum mechanical equations is one of the central problems of modern theoretical physics. In this paper an explicit semiclassical treatment of perturbation theory for non-relativistic bound states based on ℏ–­expansions and corresponding quantization conditions of one­dimensional problems is developed. The wave functions are chosen in the same way as in the logarithmic perturbation theory. Due to the introduction of new quantization conditions, this method made it possible to obtain recurrent relations, which are convenient for finding the energy eigenvalues and wave functions of bound states in the case when nodes of wave functions are taken into account. Avoiding the flaws of the standard approach, the obtained recurrent relations have the same form both for the ground and perturbed states.

Phenomenological inclusion of alternative dispersion relations to the Teukolsky equation and its application to bounding the graviton mass with gravitational-wave measurements

Existing constraints on the graviton mass from gravitational-wave detections rely on the phase difference developed between different frequencies during the propagation. Effects on the quasinormal-mode frequencies of the black-hole ringdown due to the graviton mass are often ignored. While perturbation theories of black holes have been well developed in the context of general relativity, this is not the case for modified gravity theories. We propose a phenomenological modification to the Teukolsky equation of perturbed black holes to include the dispersion relation due to a gravitational field of nonzero mass. Solving this modified Teukolsky equation by logarithmic perturbation theory, we compute the shift of the quasinormal-mode frequencies due to the presence of a graviton mass. This hypothetical shift can be used to constrain the graviton mass with ringdown signals, either standalone or in conjunction with the phase difference accumulated due to the wave propagation. We estimate that constraints on the graviton mass of $m_g \lesssim 10^{-15}\,\textrm{eV}$ can be put with a detection of the ringdown signal alone by second generation gravitational-wave detectors.

A new simple class of superpotentials in SUSY quantum mechanics

In this work, we introduce the class of quantum mechanics superpotentials W(x) = gε(x)x2n and study in detail the cases n = 0 and 1. The n = 0 superpotential is shown to lead to the known problem of two supersymmetrically related Dirac delta potentials (well and barrier). The n = 1 case results in the potentials V±(x) = g2x4 ± 2g|x|. For V−, we present the exact ground-state solution and study the excited states by a variational technique. Starting from the ground state of V− and using logarithmic perturbation theory, we study the ground states of V+ and also of V(x) = g2x4 and compare the result obtained in this new way with other results for this last potential in the literature.

Surprises in nonlinear perturbations: Case of a multiple well potential problem

AbstractWe study an atypical nonlinear perturbation problem on the lowest three states of a harmonic oscillator. The perturbation is such that the total potential is a single‐well at low values of coupling strength, turns to a double‐well in the intermediate coupling range and finally becomes a triple‐well at high values of the coupling strength. Proceeding via logarithmic perturbation theory, we notice that the perturbation furnishes exact results for both the energy and wave function of the second excited state but shows divergence for the first excited one. However, the ground state wave function is found to be exactly summable to the correct result with a finite energy series. Effects of varying the coupling constant, and hence the number of wells, on energies and wave functions of the concerned states are noted. Our results show further that the perturbed ground state acquires a pre‐exponential factor and, strangely, an exponent same as the second excited state. The origin of such an outcome is analyzed. We also highlight the reason behind the Klauder phenomenon that tells of increased sustenance of memory effects in wave functions, when compared with the Hamiltonian, in the limit of zero coupling strength. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

A perturbation theory without energy corrections

We employ logarithmic perturbation theory in a form that utilizes the force, instead of the potential, and yields equations for wavefunction-correction terms without requiring any information of energy corrections at any stage. The knowledge of unperturbed eigenfunctions of other states is also unnecessary. The perturbed energy eigenvalue can be obtained as a series either by going back to the parent equation, or as an average value involving the perturbed state. The latter scheme applies to any other average property as well. Both ground and excited states of a few systems are chosen for demonstrative calculations. Influence of the nodal structure on the exponential function controlling spatial behavior of the probability density is discussed. Interrelations among specific correction terms are shown in the small- and large-x regime; in the latter case, certain terms nicely sum up to yield the correct decay characteristics of the probability density as well. Relevance of the basic equation to an alternative, nonperturbative scheme of approximation is also outlined. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Concurrent multiple-state analytic perturbation theory via supersymmetry

Conventional nondegenerate perturbation theory for some nth state starts with the corresponding unperturbed state. The present formulation yields recursively perturbation expansions for any bound state using the sole information of the unperturbed ground state. Logarithmic perturbation theory is exploited along with supersymmetric quantum mechanics to achieve this end. As the method involves ground-state perturbations of a series of supersymmetric Hamiltonians, concern about nodal shifts of targeted excited states arises only at the ultimate step, thus, minimizing considerably the labor of clumsy computations involved in dealing with excited states.

Logarithmic perturbation theory: Applications and limitations

The time independent, non-degenerate standard perturbation theory is compared the alternate treatment of perturbation theory called logarithmic perturbation theory (LPT). For determining the non-degenerate ground state the LPT is, in principle, easier to apply than standard perturbation theory. This is because, as opposed to the standard perturbation method which requires the knowledge of the complete set of eigenvalues and eigenfunctions of the unperturbed system, for the LPT one only needs to know the ground state wave function of the unperturbed system, the energy correction to the next lower order and some easily computable coupling constants. However, in reality, the LPT is a simpler method to apply when the ground state wavefunction is exponential in nature. But as shown here for trigonometric unperturbed wave function the LPT leads to integrals which have no analytical solutions thereby making LPT more difficult and less accurate method than the standard perturbation approach. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 595-600

Logarithmic perturbation theory and some nonlinear pathological perturbations

We employ logarithmic perturbation theory to analyze selected nonlinear perturbations that are remarkable on several grounds. Our analysis offers a direct and general route to arrive at such pathological perturbing potentials for any state. Some new examples are explicitly constructed. A few issues of convergence are clarified.

Perturbative Analysis of Universality and Individuality in Gravitational Waves from Neutron Stars

The universality observed in gravitational wave spectra of non-rotating neutron stars is analyzed here. We show that the universality in the axial oscillation mode can be reproduced with a simple stellar model, namely the centrifugal barrier approximation (CBA), which captures the essence of the Tolman VII model of compact stars. Through the establishment of scaled co-ordinate logarithmic perturbation theory (SCLPT), we are able to explain and quantitatively predict such universal behavior. In addition, quasi-normal modes of individual neutron stars characterized by different equations of state can be obtained from those of CBA with SCLPT.

Logarithmic perturbation theory for a spiked oscillator and sum rules

AbstractWe show that logarithmic perturbation theory nicely yields the wavefunction correction terms in closed forms for the spiked perturbation λ/x2 on the first excited state of the harmonic oscillator, where the conventional Rayleigh‐Schrödinger scheme is known to encounter serious problems. The study also provides a direct route to obtain several sum rules, some of which appear to be new. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Generalized Supersymmetric Perturbation Theory

Using the basic ingredient of supersymmetry, a simple alternativeapproach is developed to perturbation theory in one-dimensionalnon-relativistic quantum mechanics. The formulae for the energyshifts and wavefunctions do not involve tedious calculationswhich appear in the available perturbation theories. The modelapplicable in the same form to both the ground state and excitedbound states, unlike the recently introduced supersymmetricperturbation technique which, together with other approaches basedon logarithmic perturbation theory, are involved within the moregeneral framework of the present formalism.

LOGARITHMIC PERTURBATION THEORY FOR RADIAL KLEIN–GORDON EQUATION WITH SCREENED COULOMB POTENTIALS VIA ℏ-EXPANSIONS

The explicit semiclassical treatment of logarithmic perturbation theory for the bound-state problem within the framework of the radial Klein–Gordon equation with attractive screened Coulomb potentials, contained time-component of a Lorentz four-vector and a Lorentz-scalar term, is developed. Based upon ℏ-expansions and new quantization conditions a novel procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues for the Hulthén potential containing the vector part as well as the scalar component are considered.

Logarithmic perturbation theory: A reappraisal*

AbstractWe present logarithmic perturbation theory in its simplest form that rests only on matching of certain polynomial coefficients at each order. It thus avoids any integration step in the course of calculations of energy correction terms at various orders. The scheme applies to excited states as well. Corrections to nodal point shifts are also obtained without quadratures. A few existing variants of the theory are compared and certain anomalies rectified. Merits and demerits of the approach are pointed out. We also outline a direct route to obtain perturbation series for properties through this route. This renders tedious computations of wave function correction terms at each order, obeying the corresponding normalization requirement, unnecessary, thus achieving further simplification. Pilot calculations on several model systems reveal the worth of the endeavor. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002

Equivalence of logarithmic perturbation theory and expansion of the superpotential in supersymmetric quantum mechanics

Expansion of the superpotential to get approximate solutions in supersymmetric quantum mechanics (SSQM) is shown to be equivalent to logarithmic perturbation theory (LPT). LPT formulas for energy corrections obviate cumbersome procedures in the SSQM method. Likewise, the use of SSQM partner potentials with virtually identical bound state spectra solves difficulties with excited states encountered in LPT.

A new approach to perturbation theory for a Dirac particle in a central field

The explicit semiclassical treatment of logarithmic perturbation theory for the bound-state problem within the framework of the Dirac equation is developed. Avoiding disadvantages of the standard approach in the description of exited states, new handy recursion formulae with the same simple form both for ground and exited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues for the Yukawa potential containing the vector part as well as the scalar component are considered.

Semiclassical treatment of logarithmic perturbation theory

The explicit semiclassical treatment of logarithmic perturbation theory for the nonrelativistic bound states problem is developed. Based upon -expansions and suitable quantization conditions a new procedure for deriving perturbation expansions for the one-dimensional anharmonic oscillator is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the harmonic oscillator perturbed by are considered.

Logarithmic perturbation theory for quasinormal modes

Logarithmic perturbation theory (LPT) is developed and applied to quasinormal modes (QNMs) in open systems. QNMs often do not form a complete set, so LPT is especially convenient because summation over a complete set of unperturbed states is not required. Attention is paid to potentials with exponential tails, and the example of a Poschl-Teller potential is briefly discussed. A numerical method is developed that handles the exponentially large wavefunctions which appear in dealing with QNMs.

Supersymmetry and tunneling in an asymmetric double well.

The techniques of supersymmetric quantum mechanics are applied to the calculation of the energy difference between the ground state and the first excited state of an asymmetric double well. This splitting, originating from the tunneling effect, is computed via a systematic, rapidly converging perturbation expansion. Perturbative calculations to any order can be easily carried out using a logarithmic perturbation theory. Our approach yields substantially better results than alternative widely used semiclassical analyses.

Alternative approach to nonrelativistic perturbation theory

The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.

Excited bound state logarithmic perturbation theory without nodes

The logarithmic perturbation theory is modified slightly in order to deal with excited states. Instead of considering a real wavefunction describing the physical stationary state, the authors consider a complex wavefunction at the same energy, by mixing in the ghost state. For excited bound states, the former has nodes, while the latter is guaranteed not to have any nodes, and can be represented simply as exp(-G), to which the logarithmic perturbation method can be applied in a straightforward manner. The physical entities (the energy corrections) are independent of the amount of mixing of the ghost state. The connection to the Green function method is also shown. The freedom to mix in the ghost state allows the authors to justify an ad hoc approach whereby the simple version of the logarithmic perturbation theory is applied to excited bound states. The formalism is illustrated with simple examples.