Variational improvement of perturbation theory and/or perturbative improvement of variational calculations

2008 ◽  
pp. 126-136
Author(s):  
A. Neveu
Keyword(s):  
Perturbation Theory ◽  
Variational Calculations

scholarly journals Radial anharmonic oscillator: Perturbation theory, new semiclassical expansion, approximating eigenfunctions. II. Quartic and sextic anharmonicity cases

2020 ◽  
Vol 35 (01) ◽  
pp. 2050005
Author(s):  
J. C. del Valle ◽  
A. V. Turbiner
Keyword(s):  
Perturbation Theory ◽  
Anharmonic Oscillator ◽  
Strong Coupling Regime ◽  
Weak Coupling Regime ◽  
Fractional Powers ◽  
Anharmonic Oscillators ◽  
Relative Deviation ◽  
Variational Calculations ◽  
Compact Approximation ◽  
Semiclassical Expansion

In our previous paper I (del Valle–Turbiner, 2019) a formalism was developed to study the general [Formula: see text]-dimensional radial anharmonic oscillator with potential [Formula: see text]. It was based on the Perturbation Theory (PT) in powers of [Formula: see text] (weak coupling regime) and in inverse, fractional powers of [Formula: see text] (strong coupling regime) in both [Formula: see text]-space and in [Formula: see text]-space, respectively. As a result, the Approximant was introduced — a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations, it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper, the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials [Formula: see text], [Formula: see text], respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8–12 figures for any [Formula: see text] and [Formula: see text], while the relative deviation of the Approximant from the exact eigenfunction is less than [Formula: see text] for any [Formula: see text].

Effect of Dimensionality and Anisotropy on the Holstein Polaron

1998 ◽  
Vol 543 ◽  
Author(s):  
Katja Lindenberg ◽  
A. H. Romero ◽  
D. W. Brown
Keyword(s):  
Perturbation Theory ◽  
Quantum Monte Carlo ◽  
State Energy ◽  
The Self ◽  
Monte Carlo Data ◽  
Self Trapping ◽  
Variational Calculations ◽  
Crystal Model ◽  
Polaron Radius ◽  
Mass Enhancement

AbstractWe apply weak-coupling perturbation theory and strong-coupling perturbation theory to the Holstein molecular crystal model in order to elucidate the effects of anisotropy on polaron properties in D dimensions. The ground state energy is considered as a primary criterion through which to study the effects of anisotropy on the self-trapping transition, with particular attention given to shifting of the self-trapping line and the adiabatic critical point. The effects of dimensionality and anisotropy on electron-phonon correlations and polaronic mass enhancement are studied, with particular attention given to the polaron radius and the characteristics of quasi-l-D and quasi-2-D structures. Perturbative results are confirmed by selected comparisons with variational calculations and quantum Monte Carlo data.

Lower Bounds to Expectation Values: Two-Electron Atoms

1973 ◽  
Vol 51 (2) ◽  
pp. 260-264 ◽  
Author(s):  
D. P. Chong ◽  
F. Weinhold
Keyword(s):  
Perturbation Theory ◽  
Lower Bound ◽  
Excited States ◽  
Lower Bounds ◽  
Order Perturbation ◽  
Expectation Values ◽  
High Quality ◽  
Variational Calculations ◽  
Very High ◽  
Rule Out

Weinhold's lower bound formulas are used on rather accurate wavefunctions for the 11S ground states of H−, He, and Li+, and the 21S and 23S excited states of He. The results are found to be of very high quality, as judged by comparison with the best available variational calculations, and can be used to rule out many of the values calculated by Scherr and Knight from high-order perturbation theory.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

Energy, Information, and Superluminal Speed in Quantum Electrodynamics

2020 ◽  
pp. 27-33
Author(s):  
Boris A. Veklenko
Keyword(s):  
Electromagnetic Field ◽  
Perturbation Theory ◽  
Quantum Electrodynamics ◽  
Excited Atom ◽  
Standard Quantum ◽  
Energy Information

Without using the perturbation theory, the article demonstrates a possibility of superluminal information-carrying signals in standard quantum electrodynamics using the example of scattering of quantum electromagnetic field by an excited atom.

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