diff --git "a/I9FJT4oBgHgl3EQfvy0q/content/tmp_files/load_file.txt" "b/I9FJT4oBgHgl3EQfvy0q/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/I9FJT4oBgHgl3EQfvy0q/content/tmp_files/load_file.txt" @@ -0,0 +1,1312 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf,len=1311 +page_content='Quantum information entropy of heavy mesons in the presence of a point-like defect C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Almeida1a, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Edet2b,c,d, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lima3a, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Ali4c,e, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Asjad5f aUniversidade Federal do Cear´a (UFC), Departamento de F´ısica, Campus do Pici, Fortaleza-CE, 60455-760, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' bInstitute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' cFaculty of Electronic Engineering Technology, Universiti Malaysia Perlis, Malaysia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' dDepartment of Physics, Cross River University of Technology, Calabar, Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' eAdvanced Communication Engineering (ACE) Centre of Excellence, Universiti Malaysia Perlis, 01000 Kangar, Perlis, Malaysia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' fDepartment of Mathematics, Khalifa University, Abu Dhabi 127788, United Arab Emirates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Abstract Using Schr¨odinger’s formalism, we investigate the quantum eigenstates of the heavy mesons trapped by a point-like defect and by Cornell’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' One implements this defect to the model considering a spherical metric profile coupled to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, the Nikiforov-Uvarov method is applied to theory to study the quantum eigenstates of the heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To calculate the quantum information entropy (QIE), one considers the wave functions that describe the charmonium and bottomonium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To explore the QIE, we use the well-known Shannon’s entropy formulated at the position and reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The analysis of the QIE gives us relevant information about how the quantum information change with the variation of the point-like defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Consequently, considering the Bialynicki-Birula and Mycielski (BBM) relation, we show how this defect influences the quarkonium position and momentum uncertainty measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Keywords: Schr¨odinger equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Nikiforov-Uvarov method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Cornell potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Introduction The growing interest in non-relativistic quantum mechanical systems is notorious [1, 2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This interest is because the results from quantum mechanics give us a good prediction of some phenomenological data [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, these systems are the first step toward understanding more complex models [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, considering non-relativistic quantum mechanical systems, we found several studies in the literature [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For example, one can find some studies on theoretical measurements of mass spectra of heavy mesons [11], harmonic oscillators [12, 13, 14], potential wells [16, 15], and solid-state physics problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', position-dependent mass problems) [17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Generally, depending on the type of interaction used to describe the quantum-mechanical system, there may be some difficulties in the analytical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, to bypass these difficulties, some techniques are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 1Email address: carlos@fisica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='ufc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='br 2Email address: collinsokonedet@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='com 3Email address: cleiton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='estevao@fisica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='ufc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='br 4Email address: norshamsuri@unimap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='my 5Email address: muhammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='asjad@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='ae Preprint submitted to Annals of Physics January 30, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='11627v1 [hep-ph] 27 Jan 2023 Among these techniques, there is the well-known Nikiforov-Uvarov (NU) method [3, 20, 21, 22], the Nikiforov- Uvarov Functional Analysis (NUFA) [23, 24, 25], the Laplace transformation method (LTM) [26], the asymptotic interaction method [27], the WKB approximation method [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' However, it is relevant to highlight that among the techniques mentioned, the NU method stands out [29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Indeed, the NU method stands out because it allows parameterizing Schr¨odinger’s equation [32], making it possible to find the solution for different physical potentials of interest [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, we will conveniently use the NU method in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A relevant class of particles is mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The historical framework of these particles began in 1974 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' It was in 1974 that the Brookhaven National Laboratory announced the discovery of a new particle called J [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Simultaneously, the Stanford Linear Accelerator announced the existence of another particle called ψ [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A relevant feature of these particles found is that they have similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' So has become possible to interpret these particles as a new quark (namely, a charm quark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' With the discovery of these particles, it was possible to explain the suppression of weak kaon decays that change the flavor [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Posteriorly, the fourth quark existence, the so called bottom quark, was confirmed [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Only in 1978 Cornell’s model was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In its original proposal, Cornell’s potential took into account the quark-antiquark interaction in the heavy sector [37, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Essentially the initial assumptions were that interactions arise SU(3) color gauge symmetry with flavor broken only by quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Besides, this type of interaction admitted contributions from the Coulomb interaction (induced by a gluon exchange interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, Cornell’s potential considered contributions from a phenomenological interaction of confinement, considered linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The interactions were independent of flavor and spin, thus implementing the well- known Heavy Flavor Symmetry and Heavy Quark Spin Symmetry [37, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Seeking to investigate the quarkonia eigenstates, we will consider Cornell’s potential in our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This poten- tial proves to be effective in explaining quark confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, this potential explains the mass of the quarkonium states and its relation with the angular momentum of the hadron (known as the Regge trajectory) [37, 38, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Briefly, the Cornell potential has the following form: V (r) = −4 3 αs r + σr + v0, (1) where r is the effective radius of the quarkium state, αs is the running coupling, σ is the QCD string tension, and v0 ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Note that this potential has two contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The first contribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', − 4 3 αs r ) dominates at short distances, in general for r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1fm [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This short-range contribution is known as Yukawa’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In this case, this contribution is due to the exchange of a gluon between the quark and its antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Meanwhile, the second potential contribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', σr) is the linear confinement term responsible for describing the non-perturbative QCD effects that result in color confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In this scenario, σ is the tension of the QCD string that forms when the gluonic field lines collapse in a flow tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' One estimates that the tension value of the QCD string is σ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='18GeV2 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, it is important to mention that Cornell’s potential (1) has been gaining supporters in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For example, the Cornell potential has been used in spectroscopy studies of mesons [43] and S-wave heavy quarkonia [44], among other studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Not far from this discussion, we found, in literature, some works discussing the influence of defects in quantum eigenstates [45, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Indeed, the study of topological defects in quantum mechanics problems has aroused the 2 interest of several researchers [48, 49, 50], and this interest is due to its wide application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Generally speaking, there are three classes of defects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', the cosmic string defect, the global monopole, and domain walls [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Usually, one-dimensional cosmic string defects [53] and global monopoles [54, 55, 56] apply widely to quantum mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Indeed, in this scenario, these defects were implemented, for example, in studies of harmonic oscillators [57, 58], charged particle scattering [59], and models with exotic interaction [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Considering these applications arises the question: How does the monopole-like defect influence the eigenstate of heavy mesons?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Throughout this manuscript, we will seek to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Quantum information entropy (QIE) is a useful tool in studies on quantum information and uncertainty mea- surements in quantum-mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Indeed, the QIE arises from a reinterpretation of Shannon’s information entropy [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Essentially, this reinterpretation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', the QIE, and the Bialynicki-Birula and Mycielski [62] relation, gives us a generalized uncertainty measure of Heisenberg’s principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Due to this, the QIE has gained search- light in studies on quantum-mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, one can find several works discussing the information of quantum-mechanical systems through the QIE formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For example, some applications of the QIE in the quan- tum system appear in investigations of hyperbolic potential wells [63], the Aharonov-Bohm effect [64], and effective mass problems [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Considering the applications presented, two questions naturally arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' First, how do point-like global monopole (PGM) defects change the mesons’ eigenstates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Second, how does the quantum information modifies due to this defect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' As far as we know, our work is the first to address these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, the main purpose of this paper is to study the QIE of heavy mesons in the presence of a point-like global monopole defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To reach our purpose, we organize the work as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' II, the quantum description of heavy mesons is exposed, assuming Cornell’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' III, we present a discussion and numerical results of the QIE of the Charmonium and Bottomonium states of heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' IV, the findings are announced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Quantum description of heavy mesons As discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [54], allow us to start by considering background spacetime with PGM defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Particularly, let us assume a spherically symmetric spacetime described by the line element ds2 = −dt2 + ηijdxidxj, with i, j = 1, 2, 3, (2) where the metric tensor ηij is ηij = � � � � � 1 α2 0 0 0 r2 0 0 0 r2 sin2 θ � � � � � and ηij = � � � � � α2 0 0 0 1 r2 0 0 0 1 r2 sin2 θ � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (3) Here the α parameter is related to the PGM defect, which depends on the energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Mathematically, α2 = 1 − 8πGλ2 where λ is the energy scale and G the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Indeed, several researchers have adopted the PGM defect in their studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For example, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [65], to investigate spin-0 particles in the presence of dyons and Aharonov-Bohm effect with scalar interaction, the authors adopted the 3 existence of a PGM defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, Ahmed [47] discussed the topological effects produced by this background on spin-0 particles subject to the Kratzed potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Particularly the interest in this defect is because it induces topological effects that influence the particle’s quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Motivated by this, we consider that spinless mesons are in the spacetime (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Considering the non-relativistic scenario, one writes Schr¨odinger’s equation as − ℏ2 2µ∇2 LBψ(r, t) + V (r, t)ψ(r, t) = iℏ∂ψ(r, t) ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (4) Here, µ is the particle’s mass and ∇2 LB is the Laplace-Beltrami operator defined as ∇2 LB ≡ 1 √g ∂i(√ggij∂j), (5) where g ≡ det(gij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, we will assume, in principle, that the interaction V (r, t) = V (r), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', an arbitrary central potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Assuming the Laplace-Beltrami operator (5) and the metric signature (2), we write Schr¨odinger’s equation as follows: − ℏ2 2µr2 � α2 ∂ ∂r � r2 ∂ ∂r � + 1 sin θ ∂ ∂θ � sin θ ∂ ∂θ � + 1 sin2 θ � ∂2 ∂ϕ2 �� ψ(r, θ, ϕ, t) + V (r)ψ(r, θ, ϕ, t) = iℏ∂ψ(r, θ, ϕ, t) ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Allow us to particularize our study for the case of spinless heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In this case, the interaction of the theory is V (r) = W1r − W2 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (6) This interaction is called Cornell’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Essentially, Cornell’s potential is employed to model the quarkonium interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Besides, one finds, in literature, several investigations on quantum-mechanics systems adopting Cornell’s potential, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To study the wave eigenfunctions that the theory describes, let us assume that the particular solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (6) in terms of the eigenvalues of the angular momentum operator ˆL2 are ψ(r, t) = e− iEnlt ℏ Rnl(r) r Ylm(θ, ϕ) (7) where Ylm(θ, ϕ) are spherical harmonics and Rnl(r) is the radial wave eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (7) into (6), we have that the radial part of Schr¨odinger’s equation for Cornell’s potential in the presence of the PGM defect is d2Rnl(r) dr2 + �2µEnl α2ℏ2 − 2µW1r α2ℏ2 + 2µW2 α2ℏ2r − l(l + 1) α2r2 � Rnl(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (8) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (8) is not solvable in its current form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To solve this, allow us to adopt r → x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This change of coordinates leads us to d2Rnl(x) dx2 + 2 x dRnl(x) dx + 1 x4 �2µEnl α2ℏ2 − 2µW1 α2ℏ2x + 2µW2x α2ℏ2 − l(l + 1)x2 α2 � Rnl(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (9) 4 Here let us implement an approximation scheme (AS) on the term W1 x by assuming that there is a characteristic radius r0 of the meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In this case, one obtains the expansion of W1 x in a power series around r0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', around δ ≡ 1/r0, up to the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' By setting y = x − δ around y = 0, we obtain that d2Rnl(x) dx2 + 2x x2 dRnl(x) dx + 1 x4 � −˜ϵ + ˜β1x − ˜β2x2� Rnl(x) = 0, (10) where −˜ϵ = 2µEnl ℏ2α2 − 6µW1 α2ℏ2δ , ˜β1 = 2µW2 α2ℏ2 + 6µW1 α2ℏ2δ2 , and ˜β2 = 2µW1 α2ℏ2δ3 + l(l + 1) α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (11) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (10) is the solvable form of the NU method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The major equation closely related to this method is P ′′(x) + �τ(x) σ(x)P ′(x) + �σ(x) (σ(x))2 P(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (12) Comparing to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (12) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (10), it follows that ˜τ(x) = 2x, σ(x) = x2 and ˜σ(x) = −˜ϵ + ˜β1x − ˜β2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (13) That explicitly shows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (10) satisfies the requirement of the NU approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, it is also worth noting that ˜σ(x) and σ(x) are polynomials of at most second degree, and ˜τ(x) is at most polynomials of the first degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The NU method is popular among mathematical scientists and related disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Several authors have used this method to solve problems of similar interest [29, 68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Although the method is quite popular, it will be useful to highlight some details to make our article self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For this reason, we will detail this approach in the appendix (please, verify the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Following the steps described in the appendix [Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8)], one obtains that the self-energies and eigenfunctions of our system are, respectively, Enl = 3W1 δ − α2ℏ2 8µ � � 6µW1 α2δ2ℏ2 + 2W2 α2ℏ2 n + 1 2 + � 1 4 + 2µW1 δ3α2ℏ2 + l(l+1) α2 � � 2 (14) and Rnl(r) = Nnlr ˜ β1 2 √ ˜ϵ e−rL ˜ β1 √ ˜ϵ n (2 √ ˜ϵr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (15) Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 1 and 2 expose the behavior of the radial wave eigenfunctions for the Charmonium and Bottomonium states of heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The quantum information entropy Since the seminal paper by Claude E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Shannon on the mathematical theory of communication [61], there has been a growing interest in the studies of information entropies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [16, 45, 70, 71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In quantum mechanics, part of this interest is because the quantum information entropies are related to the uncertainty measures of the quantum system [73, 74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, this entropy tells us how good the description of the quantum states of the theory is [16, 45, 71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 5 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 r R00(r) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='7 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='0 r R10(r) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='7 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='0 r R11(r) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='7 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8 (a) (b) (c) Figure 1: Radial wave eigenfunctions of heavy mesons (Charmonium) when mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='209 Gev, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6045 Gev, W1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='20 Gev, W2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='244 Gev, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='231 Gev, and ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (a) corresponds to the state n = 0 and l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' On the other hand, Fig (b) is the state n = 1 and l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The state n = 1 and l = 1 is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4 r R00(r) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='7 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='0 r R10(r) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='7 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='0 r R11(r) α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='7 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8 (a) (b) (c) Figure 2: Radial wave eigenfunctions of heavy mesons (Bottomonium) when mb = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='823 Gev, µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4115 Gev, W1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 Gev, W2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='569 Gev, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='378 Gev, and ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (a) corresponds to the state n = 0 and l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' On the other hand, Fig (b) is the state n = 1 and l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The state n = 1 and l = 1 is in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Quantum information entropies are related to uncertainty measures through the well-known Bialynicki-Birula and Mycielski (BBM) relation [62], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', Sr + Sp ≥ D (1 + lnπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (16) Here, Sr is the information entropy at position space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Sp is the information entropy at momentum space (or reciprocal space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Besides, D describes the spatial dimension of the system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', D = 1 for the one-dimensional case, D = 2 for two-dimensional one, and D = 3 for three-dimensional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In particular, the BBM relation to the quantum theory of the heavy mesons in the presence of point-like defects is Sr + Sp ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='43419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' It is interesting to mention that the Bialynicki-Birula and Mycielski relation (16) is a sophisticated version of Heisenberg’s uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, the results of quantum information entropy must respect this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For the quantum system discussed earlier (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 2), the information entropy is Sr = − ˆ 2π 0 ˆ π 0 ˆ ∞ 0 |ψ(r, t)|2 ln[|ψ(r, t)|2]r2 sin θ dr dθ dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (17) 6 Meanwhile, the information entropy at the reciprocal space is Sk = − ˆ 2π 0 ˆ π 0 ˆ ∞ 0 |ψ(k, t)|2 ln[|ψ(k, t)|]2 k2 sin θk dkr dkθ dkφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (18) The spherical wave eigenfunction at reciprocal space is ψ(k, t) = (2π)3/2 k−1/2 ∞ � l=0 (−i)l l � m=−l Ylm(kθ, kφ) ˆ ∞ 0 J 1 2 +l(krr)r3/2 dr ˆ 2π 0 ˆ π 0 ψ(r, t)Ylm(θ, φ)r2 sin θ dθ dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (19) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (19) is Hankel’s transform of the wave eigenfunction ψ(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Hankel’s transform is a transform whose kernels are Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, Hankel’s transform also is called Bessel’s transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In general, Hankel’s transform is the two-dimensional Fourier transform of a circularly symmetric function or also spherical three-dimensional functions [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Considering the definitions presented in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (17-18), we can study the quantum information entropy of heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, we begin our study by discussing the quantum information entropy of the Charmonium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Quantum information entropy for the Charmonium Quarkonia, in particle physics, are hadronic states made of a heavy quark-antiquark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For the vector case, they are the charmonium meson J/ψ (made up of a cc pair) and bottomonium meson Υ (made up of a bb pair) and their corresponding radial excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In fact, in the literature, the word quarkonium refers to the state charmonium and bottomonium [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Briefly, Charmonium is a bound state of a quark and charmed antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' We found, in the literature, several investigations discussing the Charmonium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For example, some investigations study about properties of hot gluonic plasma [78], the classification of states of mesons and their electromagnetic decays [79], and the collision of heavy ions [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Motivated by these applications, let us now study the quantum information of this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, we will now particularize our study to the quantum information entropies of the Charmonium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To inspect Charmonium’s information it is necessary to assume mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='209 Gev, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6045 Gev, W1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='20 Gev, W2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='244 Gev, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='231 Gev, and ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Using these values, we describe eigenfunctions (7) of the Charmonium eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' With the wave eigenfunctions (7), one constructs the probability densities of the theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', |ψ(r, t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Substituting the probability density in the definition of information entropy (17) comes to the numerical results of the quantum information entropy at position space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In Tab 1, we expose the numerical result of the information at the position space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To calculate the information entropy associated with the momentum of the Charmonium, the Hankel transform (19) of the eigenfunctions (7) is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Substituting the spherical wave eigenfunctions (19) in the definition (18), one obtains the numerical results of quantum information entropy at the momentum space shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The numerical results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 1 suggest that the quantum information entropy decreases at position space as the topological defect increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In other words, when the point-like global monopole defect increase, the greater the quantum information of the Charmonium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' As a consequence of this behavior, the uncertainty measures related to the position measurements of the heavy mesons will be smaller when the α parameter increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In 7 Table 1: Quantum entropy information for the heavy mesons (Charmonium state) Eigenstates (n, l, m) Parameter α Sr Sp Sr + Sp BBM relation (0,0,0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='35657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='01643 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='34014 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='43419 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='15430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='69458 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='45972 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='43419 contrast, as the point-like global monopole defect becomes large, the information entropy associated with the momentum of the mesons decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, one can conclude that when the PGM defect increase, the momentum uncertainty measures of the particles in the Charmonium eigenstate decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, we note that the BBM relation to the Charmonium state of the quarks remains valid for all eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Finally, it is possible to note that for higher energy levels, when the parameter α → 1, BBM’s relation tends to the minimum uncertainty relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', Sr + Sp ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='43419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Comments on Bottomonium Quantum Information Entropy For the Charmonium state mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='209 Gev, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6045 Gev, W1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='20 Gev, W2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='244 Gev, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='231 Gev, and ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' For the Bottomonium status, these parameters assume the following values: mb = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='823 Gev, µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4115 Gev, W1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2 Gev, W2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='569 Gev, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='378 Gev, and ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Notice that from the Charmonium state to the Bottomonium state, there is an increment of these values and the wave function profile is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, by inspection, one perceives that the quantum information entropy of the Bottomonium state has a similar behavior to the Charmonium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Indeed, the quantum information entropy Sr decreases at the position space as the parameter α (the PGM defect) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Meanwhile, the information entropy Sp increases at the momentum space when the parameter α decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' The variation of the Sr and Sp information must vary so that the BBM relation remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' These results of the quantum information entropy of the Bottomonium state (and Charmonium pre- 8 sented previously) of the quarks agree with the predictions presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [77] using the configurational entropy formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Conclusion In this work, we investigated the Charmonium and Bottomonium states that describe heavy mesons using Schr¨odinger’s theory in the presence of a PGM defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' To carry out this study, we considered a curved background coupled with the PGM defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Using the Nikiforov-Uvarov (NU) formalism, the quantum eigenstates of the heavy mesons were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Then, considering the wave functions that describe the quarkonium eigenstates, we investigate the QIE of heavy mesons using Shannon’s formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Analyzing quantum self-states that describe the heavy mesons, one notes that the radial eigenfunctions are the associated Laguerre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, to investigate the self-states, the NU method is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This method leads us to the energy spectrum shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This spectrum tells us that in the classical limit En,l→∞ ≈ 3W1/δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', the energy will depend purely on W1, the contribution of short-range of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Meanwhile, the fundamental states will depend on short- and long-range contributions of Cornell’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Be- sides, one perceives that the PGM defect alters the quantum eigenstates of the particles as it varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Indeed, when α increases, the wave function amplitude increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, the larger the α parameter, the greater the probability of finding quarkonium in regions close to the PGM defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Analyzing the QIE, we perceive that the quantum information decreases (at the position space) as the PGM defect increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' On the other hand, it increases if the contribution of the PGM defect decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In contrast, at the reciprocal space, an opposite behavior is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, it is necessary to mention that behaviors of the quantum information are consequences of the wave function profiles that describe the Charmonium and Bottomonium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' As a direct consequence, also, of this result, it is noted that the momentum’s uncertainty measures of the heavy mesons decrease as the position uncertainties increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, it is possible to notice that the BBM uncertainty relation remains valid for all energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furthermore, the BBM relation tends to a minimum value for higher energy states when α → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This occurs because, in this scenario, the PGM defect traps the heavy meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' This work has some prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Among them, a direct perspective of this work is the study of the quantum dynamics of heavy mesons in the presence of other classes of defects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', string-like or domain wall defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Another perspective is the study of the thermodynamic and statistical properties of these particles when subjected to thermal baths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' We hope to carry out these studies soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Authors Declaration Funding This research has been carried out under LRGS Grant LRGS/1/2020/UM/01/5/2 (9012-00009) Fault-tolerant Photonic Quantum States for Quantum Key Distribution provided by Ministry of Higher Education of Malaysia (MOHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 9 Conflicts of interest/Competing interest All the authors declared that there is no conflict of interest in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Acknowledgements C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Edet acknowledges eJDS (ICTP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Almeida thanks to Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), no 309553/2021-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lima is grateful to Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES), no 88887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='372425/2019-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Review of the Nikiforov-Uvarov (NU) Method Initially proposed by Nikiforov and Uvarov, the NU method seeks to solve Hypergeometric-type differential equations of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (12) [81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' One can obtain the solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (12) employing the trial wave function: P(x) = Φ(x)yn(x), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1) which reduces the differential equation (12) to a Hypergeometric-type differential equation of the form: σ(x)y′′ n(x) + τ(x)y′ n(x) + λyn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2) Allow us to highlight that the function Φ(x) is defined as Φ′(x) Φ(x) = π(x) σ(x), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3) where π(x) is a polynomial of first-degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Meanwhile, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1) is the hypergeometric function with its polynomial solution given by Rodriques relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', yn(x) = Bn ρ(x) dn dsn [σnρ(x)] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4) The term Bn is the normalization constant, and ρ(x) is known as the weight function, which in principle must satisfy the condition given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' d ds [σρ(x)] = τ(x)ρ(x) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5) where τ(x) = ˜τ(x) + 2π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Naturally, we note here that the derivative of τ(x) should be τ(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Thus, one can obtain the eigenfunctions and eigenvalues using the expression defined by π(x) and λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=', π(x) =σ′(x) − ˜τ(x) 2 ± ��σ′(x) − ˜τ(x) 2 �2 − ˜σ(x) + kσ(x) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6) and λ = k + π′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='7) Considering the discriminant of the square root [in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6)] equal to zero, we obtain the value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' In this case, the new eigenvalue equation is λ + nτ ′(x) + n(n − 1) 2 σ′′(x) = 0, with n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='8) 10 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Solutions in Detail Substituting ˜σ(x) = −˜ϵ + ˜β1x − ˜β2x2 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6), we arrived at π(x) = ± � ˜ϵ − ˜β1x + (˜β2 + k)x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1) The discriminant of the quadratic expression under the square root above is k = ˜β2 1 − 4˜β2˜ϵ 4˜ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='2) in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='1), one obtains π(x) = ± � ˜β1x 2 √ ˜ϵ − ˜ϵ √ ˜ϵ � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='3) where π′(x) = − ˜β1 2 √ ˜ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='4) Thus, we can conclude that τ(x) = 2x − ˜β1x √ ˜ϵ + ˜ϵ √ ˜ϵ and τ ′(x) = 2 − ˜β1 √ ˜ϵ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='5) which leads us to ˜β2 1 − 4˜β2˜ϵ 4˜ϵ − ˜β1 2 √ ˜ϵ = n˜β1 2 √ ˜ϵ − n2 − n (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content='6) yields the energy equation of the Cornell potential presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' (14) References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Qiang and S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Osafile, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Okon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Inyang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Goldhagen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Leong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' McCorriston, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rhoades, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rohde, Samuel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Sens, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Snyder, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Yoh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Kephart, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 39 (1977) 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Kelly, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Horne, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Losty, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rittenberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Shimada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Trippe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Wohl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Yost, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Barash-Schmidt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Bricman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Dionisi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Mazzucato, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Montanet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Crawford, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Roos, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Armstrong (Particle Data Group), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 52 (1980) S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [37] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Eichten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Gottfried, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Kinoshita, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Kogut, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lane, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 34 (1975) 369;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [38] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Eichten, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Gottfried, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Kinoshita, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Kogut, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lane, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 65 (1990) 1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [56] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Ahmed, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 12 (2022) 8794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [57] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Furtado and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Moraes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 33 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Edet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Ikot, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Okorie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Ramantswana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Horchani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 44 (1975) 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [63] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Gil-Barrera, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Carrillo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Sun and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Dong, Entropy 24 (2022) 604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [64] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Edet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Asjad, Entropy 14 (2022) 976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [65] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Oliveira and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Mello, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Scrip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 23 (2006) 5249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [66] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Kumar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Bhardwaj, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Rani and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Chand, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A 37 (2022) 2250010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [67] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Ikot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Obagboye, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Okorie, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Inyang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Amadi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Okon and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Abdel-Aty, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Plus 137 (2022) 1370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [68] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Ya¸suk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Berkdemir and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Berkdemir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 38 (2005) 6579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [69] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Onate, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Adebimpe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lukman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Adama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Okoro and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Davids, Heliyon, 4 (2018) e00977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [70] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Edet and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Ikot, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Plus 136 (2021) 432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 14 [71] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lima, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Moreira, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Machado and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Almeida, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 121 (2021) e26749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [72] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lima, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Moreira and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Almeida, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' 121 (2021) e26645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [73] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Song, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Dong, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' A 379 (2015) 1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' [74] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9FJT4oBgHgl3EQfvy0q/content/2301.11627v1.pdf'} +page_content=' -H.' metadata={'source': 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