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Quantum information entropies of the eigenstates of the Morse potential

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Ekrem Aydiner, Cenk Orta, Ramazan Sever

Quantum Physics · Submitted 24 Feb 2006 · 18 references

Pith's verdict: A small, honest numerical check of the BBM bound for Morse eigenstates, but the missing numerics methodology and an internal spectrum error keep it below the bar for peer review.

T0 · AI review

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arXiv IDquant-ph/0602203 (version 2)
Pith IDQN23ZWNU, Pith's permanent record of this paper and its review
CitationCenk Orta, Ekrem Aydiner, Ramazan Sever. "Quantum information entropies of the eigenstates of the Morse potential." arXiv:quant-ph/0602203 (2006). https://pith.science/paper/QN23ZWNU
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The paper

The paper claims

Numerical entropies of Morse-potential eigenstates satisfy the Beckner–Bialynicki-Birula–Mycielski bound.

The reading

The paper asks whether the bound states of the one-dimensional Morse potential obey the entropic uncertainty principle, not just the variance-based Heisenberg relation. For several combinations of quantum number $n$ and potential-strength parameter $\lambda$, it numerically computes the position-space Shannon entropy $S_x$ and the momentum-space Shannon entropy $S_p$, and finds every sum $S_x+S_p$ exceeds the Beckner–Bialynicki-Birula–Mycielski (BBM) lower bound $1+\ln\pi\approx2.1447$. The entropies are said to be numerically obtained, and the authors also plot entropy densities in both spaces. A sympathetic reader would take the result as a numerical confirmation that this exactly solvable molecular model respects the state-independent entropic bound, with the sum approaching saturation as the well deepens.

What carries the argument

The working objects are the exact Morse eigenfunctions $\Psi_n^\lambda(\xi)=N e^{-\xi/2}\xi^{s/2}L_n^s(\xi)$, with $\xi=2\lambda e^{-\alpha x}$, together with the Shannon entropies $S_x=-\int |\psi(x)|^2\ln|\psi(x)|^2\,dx$ and $S_p=-\int |\phi(p)|^2\ln|\phi(p)|^2\,dp$, where $\phi(p)$ is the Fourier transform of $\psi(x)$. The test is the one-dimensional BBM inequality $S_x+S_p\ge1+\ln\pi$. The numerical evaluation of these two integrals, whose integrands are plotted as entropy densities, is the part of the argument that carries the claim.

Extended reading notes

Core claim

The central claim is that for the Morse potential $V(x)=D e^{-\alpha x}(e^{-\alpha x}-2)$, the eigenstates have position and momentum information entropies satisfying $S_x+S_p\ge 1+\ln\pi$. The paper reports numerical values in its table for $n=0,1,2,3$ and several $\lambda$; the smallest listed sum is about $2.167$ for $n=0,\lambda=4$, still above the bound. The authors present this as evidence that the BBM entropic inequality holds for the Morse potential, and they note that as $\lambda$ increases (deeper well, closer to an oscillator) the entropy sum falls toward the bound.

Load-bearing premise

The claim rests on the numerical quadrature being accurate: the paper gives no grid spacing, integration limits, or convergence checks for the entropy integrals or the Fourier transform that yields $\phi(p)$, so the tabulated sums are taken on faith.

Editorial extensions

If this is right

  • Every tabulated eigenstate exhibits $S_x+S_p\ge1+\ln\pi$, so the BBM entropic uncertainty bound is numerically respected for the Morse potential.
  • As $\lambda$ grows, the entropy sum approaches the lower bound, consistent with the Morse potential becoming more oscillator-like.
  • The position entropy density is asymmetric while the momentum entropy density is symmetric, showing localization features that variance-based uncertainty does not capture.
  • The table provides a concrete data set that later analytic or numerical work on Morse entropies can benchmark against.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: because the BBM inequality is an established theorem, the paper's real burden is numerical; an independent implementation with stated tolerances would make the table a stronger benchmark.
  • Inference: the same numerical procedure applied to other exactly solvable one-dimensional potentials, such as Pöschl–Teller, would show whether the monotone saturation toward $1+\ln\pi$ is specific to the Morse well or generic.
  • Inference: the reported trend suggests a scaling relation for the excess entropy $S_x+S_p-(1+\ln\pi)$ as a function of $\lambda$, which the paper does not extract; fitting high-precision values could reveal whether the approach is algebraic or exponential.
Original abstract

The position and momentum space information entropies for the Morse potential are numerically obtained for different strengths of the potential. It is found to satisfy the bound obtained by Beckner, Bialynicki-Birula, and Mycielski. Interesting features of the entropy densities are graphically demonstrated.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,855 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.59 · compares with prior results 0.43 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

The position and momentum space information entropies for the Morse potential are numerically obtained for different strengths of the potential. It is found to satisfy the bound obtained by Beckner, Bialynicki-Birula, and Mycielski. Interesting features of the entropy densities are graphically demonstrated.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The position and momentum space information entropies for the Morse potential are numerically obtained for different strengths of the potential. contribution 0.58 0.93 0.11 asserted 1.00
s1 It is found to satisfy the bound obtained by Beckner, Bialynicki-Birula, and Mycielski. result 0.99 0.85 0.48 asserted 1.00
s2 Interesting features of the entropy densities are graphically demonstrated. result 0.60 0.88 0.08 asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Plots of the position space entropy densities of the M
Figure 2. FIG. 2: Plots of the momentum space entropy densities of the M

Figures reproduced from arXiv:quant-ph/0602203 by the authors.

References
How the literature uses it

2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Uses its method or tool 1 paper
Cites it as background 1 paper
  • The Laguerre polynomials are known to play a crucial role not only in applied and computational mathematics [ref] , mathematical physics [ref] , combinatorics [ref] , information theory [ref] , quantum algebras [ref] , asymptotics [ref] and theory of special functions [ref] , but also in non-relativisitic, relativistic and supersymmetric quantum mechanics [ref] , atomic and molecular physics [ref] and D-dimensional physics [ref] .

    Direct spreading measures of Laguerre polynomials · Introduction · 0.87

Discussion

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