Journal ArticleDOI
On the sign of the second-order energy shift in the Rayleigh–Schrödinger perturbation theory for a highly excited state
TLDR
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.read more
Citations
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Continued Roots, Power Transform and Critical Properties
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.
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Off-diagonal matrix elements: a modus tollens approach
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.
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Restrictions on second order response properties of quantum states
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.
References
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Journal ArticleDOI
Quantum-mechanical perturbation theory
TL;DR: In this paper, the convergence properties of perturbation series and Pade approximants are reviewed in the context of traditional results from the theory of real and complex variables, and the two major versions of time-independent perturbations theory, the Rayleigh-Schrodinger (RS) and Brillouin-Wigner (BW) theories, are compared and contrasted, and alternative techniques for evaluating the terms in the energy series are discussed.
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Hypervirial and hellmann--feynman theorems applied to anharmonic oscillators.
TL;DR: In this article, a set of hypervirial theorems plus the Hellman-Feynman theorem are applied to a general anharmonic oscillator and the exact energy and expection values of powers of the position coordinate are expanded in a power series of the coupling constant.
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Transformation of perturbation series into continued fractions, with application to an anharmonic oscillator
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TOPICAL REVIEW: The Bloch wave operator: generalizations and applications: Part I. The time-independent case
Journal ArticleDOI
Alternative quantum perturbation theory without divergences
TL;DR: In this article, a different integral form of the Schrodinger equation and some boundedness conditions of the wave functions are proved, which exhibit the origin of some divergences in quantum mechanics.