On the nonlinear manifold energy variation method and excited state calculations

Pathak, R.K.; Bhattacharyya, Krishnendu

doi:10.1016/0009-2614(94)01179-6

Cited by 16 publications

(9 citation statements)

References 12 publications

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“…In this approach [70], instead of minimizing a particular energy state, trace of the matrix with respect to non-linear parameter, γ is minimized, which remains invariant under diagonalization of the matrix; thus one diagonalizes the matrix at that particular value of γ for which trace becomes minimum. As a result, one obtains all desired eigenstates in a single diagonalization step.…”

Section: Variation-induced Exact Diagonalizationmentioning

confidence: 99%

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

Mukherjee

Roy

2015

Annalen der Physik

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Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double-well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation-induced exact diagonalization procedure. The information entropy-based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase-space description provides a semiclassical explanation for this feature. Total information entropy and phase-space area show similar behavior with increasing barrier height.

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Section: Variation-induced Exact Diagonalizationmentioning

confidence: 99%

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

Mukherjee

Roy

2015

Annalen der Physik

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“…Diagonalization of the symmetric matrix, h was accomplished efficiently by MATHEMATICA, leading to accurate energy eigenvalues and corresponding eigenvectors. We adopt a Manifold‐Energy minimization approach due to , where instead of minimizing a particular energy state, one minimizes trace of the matrix, which is given below,

\begin{matrix} true \\ right T r [h] = \sum_{l} h_{l l} & = & left \sum_{l} ([, \frac{3 α}{16 σ^{2}}, (2 l^{2} + 2 l + 1)) \\ left (] - \frac{(β + 4 σ^{2}) (2 l + 1)}{4 σ} + 2 σ (2 l + 1)) . \end{matrix}

with respect to σ. This leads to a cubic equation in σ having a single real root.…”

Section: Methodsmentioning

confidence: 99%

Quantum confinement in an asymmetric double‐well potential through energy analysis and information entropic measure

Mukherjee¹,

Roy

2015

Annalen der Physik

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Localization of a particle in the wells of an asymmetric double-well (DW) potential is investigated here. Information entropy-based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phasespace area, are utilized to explain the contrasting effect of localization-delocalization and role of asymmetric term in such two-well potentials. In asymmetric situation, two wells behaves like two different potentials. A general rule has been proposed for arrangement of quasi-degenerate pairs, in terms of asymmetry parameter. Further, it enables to describe the distribution of particle in either of the deeper or shallow wells in various energy states. One finds that, all states eventually get localized to the deeper well, provided the asymmetry parameter attains certain threshold value. This generalization produces symmetric DW as a natural consequence of asymmetric DW. Eigenfunctions, eigenvalues are obtained by means of a simple, accurate variationinduced exact diagonalization method. In brief, information measures and phase-space analysis can provide valuable insight toward the understanding of such potentials.

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“…Hence, we may define the control function s as that providing the minimum of the multiplier. Instead of the multiplier (13), as a function of the variable f, it may be more convenient to pass to its image…”

Section: Algebraic Transformsmentioning

confidence: 99%

“…This implies that the multiplier µ k → 0, as k → ∞. Another quantity related to the multiplier (13) is the predictability time [35] which can be defined as τ k ≈ |λ k | −1 , or τ k ≈ |k/ ln |µ k || . This is the characteristic time during which the motion along the cascade trajectory effectively approaches a fixed point.…”

Section: Algebraic Transformsmentioning

confidence: 99%

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Self-similar bootstrap of divergent series

1997

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A method is developed for calculating effective sums of divergent series. This approach is a variant of the self-similar approximation theory. The novelty here is in using an algebraic transformation with a power providing the maximal stability of the self-similar renormalization procedure. The latter is to be repeated as many times as it is necessary in order to convert into closed self-similar expressions all sums from the series considered. This multiple and complete renormalization is called self-similar bootstrap. The method is illustrated by several examples from statistical physics.64.60. Ak, 11.10.Gh, 64.10+h, 02.30.Lt

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On the nonlinear manifold energy variation method and excited state calculations

Cited by 16 publications

References 12 publications

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

Information entropy as a measure of tunneling and quantum confinement in a symmetric double‐well potential

Quantum confinement in an asymmetric double‐well potential through energy analysis and information entropic measure

Self-similar bootstrap of divergent series

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