{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T12:56:25Z","timestamp":1774270585323,"version":"3.50.1"},"reference-count":65,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T00:00:00Z","timestamp":1774137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we solve the Schr\u00f6dinger equation for a hydrogenic impurity located at the apex of a right circular cone, with the electron constrained to move on the conical surface of semi-aperture angle \u03b80 and subjected to an Aharonov\u2013Bohm magnetic flux along the symmetry axis. Analytical expressions for the energy eigenvalues and normalized radial wave functions are obtained in terms of the principal quantum number n and the angular quantum number m, the magnetic flux \u03bd, and the cone angle. The Shannon entropy is evaluated in both configuration and momentum spaces for several low-lying states, and its variation with \u03bd and \u03b80 is analyzed in detail. When the magnetic flux vanishes, pairs of states n,\u00a0m and n,\u00a0\u2212m share the same entropic behavior; for finite flux, this degeneracy is lifted and the entropies depend explicitly on the state, the cone geometry, and the flux strength. Finally, we verify that the entropic sum Sr+Sp fulfills the Bialynicki-Birula\u2013Mycielski bound, providing an information-theoretic consistency check for the model.<\/jats:p>","DOI":"10.3390\/e28030356","type":"journal-article","created":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T11:59:36Z","timestamp":1774267176000},"page":"356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Shannon Entropy of a Hydrogenic Impurity on a Conical Surface: Confinement and Aharonov\u2013Bohm Effects"],"prefix":"10.3390","volume":"28","author":[{"given":"Luis Manuel","family":"Arvizu","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidad Aut\u00f3noma Metropolitana-Iztapalapa, Av. Ferrocarril de San Rafael Atlixco 186, Col. Leyes de Reforma 1\u00aa Secci\u00f3n, Iztapalapa, Mexico City 09310, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2951-5406","authenticated-orcid":false,"given":"Eleuterio","family":"Casta\u00f1o","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidad Aut\u00f3noma Metropolitana-Iztapalapa, Av. Ferrocarril de San Rafael Atlixco 186, Col. Leyes de Reforma 1\u00aa Secci\u00f3n, Iztapalapa, Mexico City 09310, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Norberto","family":"Aquino","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidad Aut\u00f3noma Metropolitana-Iztapalapa, Av. Ferrocarril de San Rafael Atlixco 186, Col. Leyes de Reforma 1\u00aa Secci\u00f3n, Iztapalapa, Mexico City 09310, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Frieden, B.R. (1998). Physics from Fisher Information: A Unification, Cambridge University Press.","DOI":"10.1017\/CBO9780511622670"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_3","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, John Wiley & Sons, Inc.. [2nd ed.]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0921-8890(99)00002-0","article-title":"A model-based sound localization system and its application to robot navigation","volume":"27","author":"Huang","year":"1999","journal-title":"Robot. Auton. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1080\/00036847200000017","article-title":"Entropy, market risk, and the selection of efficient portfolios","volume":"4","author":"Philippatos","year":"1972","journal-title":"Appl. Econ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1109\/TFUZZ.2008.924200","article-title":"Mean-entropy models for fuzzy portfolio selection","volume":"16","author":"Huang","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Yang, L. (2012). Study on cumulative residual entropy and variances as risk measure. 2012 Fifth International Conference on Business Intelligence and Financial Engineering, IEEE.","DOI":"10.1109\/BIFE.2012.52"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1109\/TKDE.2011.261","article-title":"Information-theoretic outlier detection for large-scale categorical data","volume":"25","author":"Wu","year":"2011","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1080\/24725854.2017.1299952","article-title":"Detecting entropy increase in categorical data using maximum entropy distribution approximations","volume":"49","author":"Das","year":"2017","journal-title":"IISE Trans."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"012085","DOI":"10.1088\/1742-6596\/135\/1\/012085","article-title":"Inverse problem in machine learning: An application to brain activity interpretation","volume":"135","author":"Prato","year":"2008","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/0031-3203(81)90094-7","article-title":"Pattern recognition as a quest for minimum entropy","volume":"13","author":"Watanabe","year":"1981","journal-title":"Pattern Recognit."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Llenas, A., Riihijarvi, J., and Petrova, M. (2017). Performance evaluation of machine learning based signal classification using statistcal and multiscale entropy features. 2017 IEEE Wireless Comunication and Networking Conference (WCNC), IEEE.","DOI":"10.1109\/WCNC.2017.7925865"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Shwartz, R., and LeCun, Y. (2024). To compress or Not to compress-Self-supervised Learning and information theory: A Review. Entropy, 26.","DOI":"10.3390\/e26030252"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Wang, Y. (2016). Seismic Inversion: Theory and Applications, Wiley Blackwell.","DOI":"10.1002\/9781119258032"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1049\/ip-f-1.1984.0099","article-title":"Maximum entropy method in image processing","volume":"Volume 131","author":"Gull","year":"1984","journal-title":"IEE Proceedings F (Communications, Radar and Signal Processing)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/j.ins.2012.07.049","article-title":"Local Shannon entropy measure with statistical test for image randomnes","volume":"222","author":"Wu","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_17","first-page":"16","article-title":"On the role of Tsallis entropy in image processing","volume":"1","author":"Sparavigna","year":"2015","journal-title":"Int. Sci. Res. J."},{"key":"ref_18","unstructured":"Bertero, M., and Plana, M. (2006). Inverse Problems in Biomedical Imaging: Modeing and Methods of Solution, Springer."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s11071-011-0281-2","article-title":"Entropy measures for biological signal analyses","volume":"68","author":"Gao","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"047905","DOI":"10.1103\/PhysRevLett.92.047905","article-title":"Continuous-Variable Quantum Cryptography is Secure against Non-Gaussian Attacks","volume":"92","author":"Grasshans","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"e2310644","DOI":"10.1080\/00268976.2024.2310644","article-title":"On information, entropy and early stone tools","volume":"122","author":"Chaparro","year":"2024","journal-title":"Mol. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"075014","DOI":"10.1088\/1361-6382\/acbe89","article-title":"Configurational entropy of black hole quantum cores","volume":"40","author":"Casadio","year":"2023","journal-title":"Class. Quantum Grav."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"e25977","DOI":"10.1002\/qua.25977","article-title":"The Shannon entropy of high-dimensional hydrogenic and harmonic systems","volume":"119","author":"Dehesa","year":"2019","journal-title":"Int. J. Quantum Chem."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2215","DOI":"10.1016\/j.physa.2011.02.020","article-title":"Entropy and complexity analysis of Dirac-delta-like quantum potentials","volume":"390","author":"Bouvrie","year":"2011","journal-title":"Phys. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"050302","DOI":"10.1088\/1674-1056\/22\/5\/050302","article-title":"Quantum information entropies of the eigenstates for the P\u00f6schl-Teller-like potential","volume":"22","author":"Sun","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4512","DOI":"10.1103\/PhysRevA.57.4512","article-title":"Calculating the logarithmic mean excitation energy from the Shannon information entropy of the electronic charge density","volume":"57","author":"Weaver","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.cplett.2004.03.118","article-title":"Order of Coulomb and Fermi pairs: Application in a \u03c0-system","volume":"389","author":"Karafiloglou","year":"2004","journal-title":"Chem. Phys. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1002\/qua.560190106","article-title":"Molecular properties and basis set quality: An approach based on information theory","volume":"19","author":"Maroulis","year":"1981","journal-title":"Int. J. Quantum Chem."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1002\/qua.560240603","article-title":"Basis set quality. II. Information theoretic appraisal of various s-orbitals","volume":"24","author":"Simas","year":"1983","journal-title":"Int. J. Quantum Chem."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2602","DOI":"10.1103\/PhysRevA.32.2602","article-title":"Some novel characteristics of atomic information entropies","volume":"32","author":"Grade","year":"1985","journal-title":"Phys. Rev. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3842","DOI":"10.1021\/cr030088+","article-title":"Nucleus-independent chemical shifts (NICS) as an aromaticity criterion","volume":"105","author":"Chen","year":"2005","journal-title":"Chem. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0009-2614(94)00029-8","article-title":"A numerical study of molecular information entropies","volume":"219","author":"Sagar","year":"1994","journal-title":"Chem. Phys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1002\/(SICI)1097-461X(1996)58:4<323::AID-QUA1>3.0.CO;2-#","article-title":"Information entropy as a measure of the quality of an approximate electronic wave function","volume":"58","author":"Nagy","year":"1996","journal-title":"Int. J. Quantum Chem."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"129826","DOI":"10.1016\/j.physa.2024.129826","article-title":"Quantum entropic exchangeat avoided crossing due to laser-atom interaction","volume":"643","author":"Prasad","year":"2024","journal-title":"Physica A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2401194","DOI":"10.1002\/adts.202401194","article-title":"Analysis of Shannon entropy and Quantum states of a confined hydrogen atom screened by the Hellmann potential","volume":"8","author":"Kumar","year":"2025","journal-title":"Adv. Theory Simul."},{"key":"ref_36","first-page":"060401","article-title":"Shannon entropy along hydrogen isoelectric sequence using numerov method","volume":"69","author":"Joshi","year":"2023","journal-title":"Rev. M\u00e9x. F\u00eds."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"01016","DOI":"10.1051\/epjconf\/202024401016","article-title":"Classification of isentropic molecules in terms of Shannon entropy","volume":"244","author":"Savirov","year":"2020","journal-title":"EPJ Web Conf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"074110","DOI":"10.1063\/1.2008212","article-title":"Characteristic features of Shannon information entropy of confined atoms","volume":"123","author":"Sen","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"208","DOI":"10.3390\/quantum1020018","article-title":"Shannon entropy for the hydrogen atom confined by four different potentials","volume":"1","author":"Vargas","year":"2019","journal-title":"Quantum Rep."},{"key":"ref_40","first-page":"e25571","article-title":"Electron-density delocalization in many-electron atoms confined by penetrable walls: A Hartree\u2013Fock study","volume":"118","author":"Vargas","year":"2017","journal-title":"Int. J. Quantum Chem."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"126988","DOI":"10.1016\/j.physleta.2020.126988","article-title":"Shannon entropy and Fisher information for endohedral confined one and two-electron atoms","volume":"386","year":"2021","journal-title":"Phys. Lett. A"},{"key":"ref_42","first-page":"757","article-title":"Sobre um estudo da entropia de Shannon no contexto da mec\u00e2nica qu\u00e2ntica: Uma aplica\u00e7\u00e3o ao oscilador harm\u00f4nico livre e confinado","volume":"39","author":"Nascimento","year":"2016","journal-title":"Quim. Nova"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s10910-012-0073-z","article-title":"Measuring localization-delocalization phenomena in a quantum corral","volume":"51","author":"Corzo","year":"2013","journal-title":"J. Math. Chem."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1402","DOI":"10.1016\/j.physleta.2015.03.020","article-title":"Shannon information entropy for an infinite circular well","volume":"379","author":"Song","year":"2015","journal-title":"Phys. Lett. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"085107","DOI":"10.1088\/1402-4896\/ace2f3","article-title":"Quantum-information theory of magnetic field influence on circular dots with different boundary conditions","volume":"98","author":"Shafeekali","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"e27358","DOI":"10.1002\/qua.27358","article-title":"The confined helium atom: An informational-theoretic approach","volume":"124","author":"Montgomery","year":"2024","journal-title":"Int. J. Quantum Chem."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"189","DOI":"10.3390\/quantum2010012","article-title":"Shannon entropy in confined helium-like ions within a Density Functional formalism","volume":"2","author":"Majumdar","year":"2020","journal-title":"Quantum Rep."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"095401","DOI":"10.1088\/1402-4896\/abaa09","article-title":"R\u00e9nyi, Fisher, Shannon and their electron correlation tools for two-electron series","volume":"95","author":"Nasser","year":"2020","journal-title":"Phys. Scr."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1139\/cjp-2019-0391","article-title":"Fisher information and Shannon entropy calculations for two-electron systems","volume":"98","author":"Nasser","year":"2020","journal-title":"Can. J. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"145301","DOI":"10.1088\/1751-8121\/adc4a3","article-title":"Shannon entropy of an electron on a conical surface: Aharonov-Bohm effects","volume":"58","author":"Arvizu","year":"2025","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"095270","DOI":"10.1088\/1402-4896\/ad6fe0","article-title":"Shannon entropy of a particle on a conical surface","volume":"99","author":"Arvizu","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1110","DOI":"10.1016\/j.physleta.2018.12.040","article-title":"Quantum information measures of the Aharonov-Bohm ring in uniform magnetic fields","volume":"383","author":"Olendski","year":"2019","journal-title":"Phys. Lett. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1140\/epjp\/s13360-022-02627-5","article-title":"Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field","volume":"137","author":"Olendski","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1140\/epjd\/s10053-023-00718-1","article-title":"Information-theoretic measures and thermodynamic properties under magnetic and Aharonov-Bohm flux fields","volume":"77","author":"Omugbe","year":"2023","journal-title":"Eur. Phys. J. D"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"065111","DOI":"10.1088\/1402-4896\/acd309","article-title":"Quantum information entropy of a particle trapped by the Aharonov-Bohm-type effect","volume":"98","author":"Lima","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"114440","DOI":"10.1016\/j.physe.2020.114440","article-title":"Effects of the external electric and magnetic fields of the linear and nonlinear optical properties of InAs cylindrical quantum dot with modified Poschl-Teller and Morse confinement potentials","volume":"126","author":"Sargsian","year":"2021","journal-title":"Phys. E Low-Dims. Syst. Nanostruct."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"094309","DOI":"10.1063\/1.3582137","article-title":"The electric field effects on binding energies and nonlinear optical properties of donor impurity in spherical quantum dot","volume":"109","author":"Kirak","year":"2011","journal-title":"J. Appl. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1313","DOI":"10.1002\/pssb.200844349","article-title":"Absorption spectra of a donor impurity in quantum ring","volume":"246","author":"Xie","year":"2009","journal-title":"Phys. Status Solidi"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"4588","DOI":"10.1016\/j.physb.2012.08.023","article-title":"Optical anisotropy of donor in ellipsoidal dots","volume":"407","author":"Xie","year":"2012","journal-title":"Phys. B Condens. Matter"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.optcom.2016.03.064","article-title":"Magneto-absorption in conical quantum dot ensemble: Possible applications for QD LED","volume":"371","author":"Hayrapetyan","year":"2016","journal-title":"Opt. Commun."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"2335","DOI":"10.1166\/jctn.2011.1965","article-title":"The analytical study of electronic and optical properties of pyramid-like and cone-like quantum dots","volume":"8","author":"Lozovski","year":"2011","journal-title":"J. Comput. Theor. Nanosci."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"025401","DOI":"10.1088\/1402-4896\/ae31b3","article-title":"Hydrogenic atom on the surface of a cone","volume":"101","author":"Aquino","year":"2026","journal-title":"Phys. Scr."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1186","DOI":"10.1103\/PhysRevA.43.1186","article-title":"Analytical solution of a two-dimensional hydrogen atom. I Nonrelativistic theory","volume":"43","author":"Yang","year":"1991","journal-title":"Phys. Rev. A"},{"key":"ref_64","first-page":"628","article-title":"Confinement effects in hydrogen-like two-dimensional atoms","volume":"46","author":"Aquino","year":"1998","journal-title":"Rev. Mex. F\u00eds."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF01608825","article-title":"Uncertainty relations for information entropy in wave mechanics","volume":"44","author":"Mycielski","year":"1975","journal-title":"Commun. Math. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/356\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T12:04:34Z","timestamp":1774267474000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/356"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,22]]},"references-count":65,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["e28030356"],"URL":"https:\/\/doi.org\/10.3390\/e28030356","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,22]]}}}