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Journal ArticleDOI

On the sign of the second-order energy shift in the Rayleigh–Schrödinger perturbation theory for a highly excited state

TL;DR: The second-order energy shift in the Rayleigh-Schrodinger perturbation theory is most likely to be positive for large quantum numbers, i.e. for excited states.
Abstract: The second-order energy shift in the Rayleigh–Schrodinger perturbation theory is most likely to be positive for large quantum numbers, i.e. for excited states. Within the Wilson–Sommerfeld approximation, good for large quantum numbers, we show on the contrary that the sufficient condition for the second-order shift to be negative is, that the total potential be symmetric and both the unperturbed and perturbed potentials be monotonically increasing.
Citations
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Journal ArticleDOI
19 Aug 2021-Symmetry
TL;DR: In this paper, the authors considered the problem of computing the critical amplitudes at infinity by means of the self-similar continued root approximants, which can be found from the optimization imposed on the parameters of power transform.
Abstract: We consider the problem of calculation of the critical amplitudes at infinity by means of the self-similar continued root approximants. Region of applicability of the continued root approximants is extended from the determinate (convergent) problem with well-defined conditions studied before by Gluzman and Yukalov (Phys. Lett. A 377 2012, 124), to the indeterminate (divergent) problem my means of power transformation. Most challenging indeterminate for the continued roots problems of calculating critical amplitudes, can be successfully attacked by performing proper power transformation to be found from the optimization imposed on the parameters of power transform. The self-similar continued roots were derived by systematically applying the algebraic self-similar renormalization to each and every level of interactions with their strength increasing, while the algebraic renormalization follows from the fundamental symmetry principle of functional self-similarity, realized constructively in the space of approximations. Our approach to the solution of the indeterminate problem is to replace it with the determinate problem, but with some unknown control parameter b in place of the known critical index β. From optimization conditions b is found in the way making the problem determinate and convergent. The index β is hidden under the carpet and replaced by b. The idea is applied to various, mostly quantum-mechanical problems. In particular, the method allows us to solve the problem of Bose-Einstein condensation temperature with good accuracy.

4 citations

Journal ArticleDOI
TL;DR: In this article, a numerical application of the modus tollens rule of classical logic is described and is shown to permit the calculation of off-diagonal matrix elements by using a small basis set of implicit wavefunctions.
Abstract: A numerical application of the modus tollens rule of classical logic is described and is shown to permit the calculation of off-diagonal matrix elements by using a small basis set of implicit wavefunctions. The success of the calculation is discussed in terms of the most simple form of the Hohenberg?Kohn theorem.
Journal ArticleDOI
TL;DR: In this paper, the authors distinguish two extreme classes of perturbation problems depending on the signs of second-order response properties of quantum states, i.e., positive values of the same for any state, and overwhelmingly more probable for highly excited states.
Abstract: We distinguish two extreme classes of perturbation problems depending on the signs of second-order response properties. The first class refers to a positive value of the same for any state, and is overwhelmingly more probable. The other category offers all-but-one negative values, or at least some negative values for highly excited states. The classes are seen to differ in reproducing results of finite-dimensional matrix Hamiltonian perturbations, allowing the emergence of a type of sum rule. A few analytical findings are employed for direct demonstration. The outcomes provide notable restrictions on second order response properties of quantum states.
References
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Book
01 Jan 1961

20,079 citations

Book
01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10 The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field Zentralblatt MATH, 836

19,846 citations

Journal ArticleDOI
TL;DR: In this article, the authors review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications, including shape invariance and operator transformations, and show that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials.

2,688 citations

Book ChapterDOI
TL;DR: Perturbation theory is designed to deal systematically with the effects of small perturbations on physical systems when the effects are mathematically too difficult to calculate exactly, and the properties of the unperturbed system are known as mentioned in this paper.
Abstract: Publisher Summary The purpose of this chapter is to provide information on the recent developments in perturbation theory. In recent years, there is a great increase of interest in the application of perturbation theory to the fundamental problems of quantum chemistry. Perturbation theory is designed to deal systematically with the effects of small perturbations on physical systems when the effects of the perturbations are mathematically too difficult to calculate exactly, and the properties of the unperturbed system are known. The new applications have been mainly to atoms where the reciprocal of the atomic number, l/Z, provides a natural perturbation parameter. These may be divided into two groups. The first consists of calculations of energy levels, and is a natural outgrowth of Hylleraas's classic work on the 1/Z expansion for two-electron atoms. The applications in the second group are to the calculation of expectation values and other properties of atoms and molecules, and are of much more recent origin. There are two principal reasons for the success of these new applications: (1) sufficient accuracy is frequently obtained from knowledge of a first-order perturbed wave function, and (2) a great advantage of perturbation theory is that the functional form of the perturbed wave function is shaped by the perturbation itself.

551 citations

Journal ArticleDOI
TL;DR: In this article, hypervirial relations are used to yield the series for the energy and other expectation values, for a hydrogen atom with perturbation λ r, without calculation of perturbed wavefunctions.

172 citations