Compact Equations for the Envelope Theory

Cimino, Lorenzo; Semay, Claude

doi:10.1007/s13538-021-01047-7

Cited by 4 publications

(4 citation statements)

References 22 publications

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“…For some Hamiltonians, the ET presents a variational character, guiding through the exact result. Recently, the ET and its improvement have been generalised for systems of N a identical particles of type a and one particle of type b [16,22]. Most tests performed with this generalisation proved to be conclusive but the study of atomic spectra revealed an unexpected low accuracy, even after improvement [24].…”

Section: Discussionmentioning

confidence: 99%

“…The approximation method can deal with Hamiltonian including one-body potentials and some kind of K-body forces [20]. Furthermore, the method has also been recently generalised to systems of N a identical particles of type a and N b identical particles of type b [16,22]. Both generalisations lead to a new set of compact equations ( 2) but the rest of the discussion remains relevant.…”

Section: Variational Charactermentioning

confidence: 99%

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Study on the Accuracy of the Envelope Theory

Cimino¹,

Chevalier²,

Carlier³

et al. 2023

Preprint

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The envelope theory is an easy-to-use approximation method to obtain eigensolutions for some quantum many-body systems. Even if the solutions are reliable and an improvement procedure exists, the method can lack of accuracy for some systems. In a previous work, two hypotheses were proposed to explain the low precision: the presence of a divergence in the potential and a mix of variational characters. In the present work, different systems are studied to test these hypotheses. Theses tests show that the presence of a divergence causes indeed less accurate results, while the mix of variational characters reduces the impact of the improvement procedure.

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Section: Discussionmentioning

confidence: 99%

Section: Variational Charactermentioning

confidence: 99%

Study on the Accuracy of the Envelope Theory

Cimino¹,

Chevalier²,

Carlier³

et al. 2023

Preprint

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“…In this section, matters from several papers [5,10,11] are collected and summarised to describe how to compute the approximate ET solutions for the generic one-dimensional Hamiltonian given by…”

Section: The Envelope Theorymentioning

confidence: 99%

“…In this last reference, a summary of the calculations performed in [13] gives the main steps for the construction of the ET. More recently, the compact equations for two sets of different particles have also been computed [11].…”

Section: The Envelope Theorymentioning

confidence: 99%

The envelope theory as a pedagogical tool

Semay

Maud

2023

Eur. J. Phys.

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The envelope theory is a reliable and easy to implement method to solve time independent Schr"odinger-like equations (eigenvalues and eigenvectors). It is particularly useful to solve many-body systems since the computational cost is independent from the number of particles. The purpose of this paper is twofold. First, we want to make known a method that is probably too little used. Second, we also want to show that this method can be used as a pedagogical tool, thanks to its simplicity and the reliable results that can be obtained. To reach these goals, the envelope theory is applied to a simple problem in one dimension, the soft-Coulomb potential $-k/\sqrt{x^2+d^2}$, characterised by a bias distance $d$. Such interaction is used for the study of excitons, electron-hole bound pairs where the two charges are kept separated in two different one-dimensional regions (quantum wires). In addition to its physical interest, this system has never been treated with the envelope theory.

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