{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:26:23Z","timestamp":1760235983430,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T00:00:00Z","timestamp":1634169600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Agencia Estatal de Investigacion (Spain)","award":["PID2020-113390GB-I00"],"award-info":[{"award-number":["PID2020-113390GB-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>High dimensional atomic states play a relevant role in a broad range of quantum fields, ranging from atomic and molecular physics to quantum technologies. The D-dimensional hydrogenic system (i.e., a negatively-charged particle moving around a positively charged core under a Coulomb-like potential) is the main prototype of the physics of multidimensional quantum systems. In this work, we review the leading terms of the Heisenberg-like (radial expectation values) and entropy-like (R\u00e9nyi, Shannon) uncertainty measures of this system at the limit of high D. They are given in a simple compact way in terms of the space dimensionality, the Coulomb strength and the state\u2019s hyperquantum numbers. The associated multidimensional position\u2013momentum uncertainty relations are also revised and compared with those of other relevant systems.<\/jats:p>","DOI":"10.3390\/e23101339","type":"journal-article","created":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T09:10:49Z","timestamp":1634202649000},"page":"1339","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4397-9426","authenticated-orcid":false,"given":"Jes\u00fas S.","family":"Dehesa","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica At\u00f3mica, Molecular y Nuclear, Universidad de Granada, 18071 Granada, Spain"},{"name":"Instituto Carlos I de F\u00edsica Te\u00f3rica y Computacional, Universidad de Granada, 18071 Granada, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1103\/PhysRevA.11.42","article-title":"Variable dimensionality in atoms and its effect on the ground state of the helium isoelectronic sequence","volume":"11","author":"Herrick","year":"1975","journal-title":"Phys. Rev. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1063\/1.2914163","article-title":"Quarks, atoms, and the 1\/N expansion","volume":"33","author":"Witten","year":"1980","journal-title":"Phys. Today"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1063\/1.450584","article-title":"Dimensional interpolation for two-electron atoms","volume":"84","author":"Herschbach","year":"1986","journal-title":"J. Chem. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Herschbach, D.R., Avery, S., and Goscinski, O. (1992). Dimensional Scaling in Chemical Physics, Kluwer.","DOI":"10.1007\/978-94-011-1836-1"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1007\/s10910-010-9710-6","article-title":"Variational justification of the dimensional-scaling method in chemical physics: The H-atom","volume":"48","author":"Chen","year":"2010","journal-title":"J. Math. Chem."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1080\/01442350802364664","article-title":"Bohr model and dimensional scaling analysis of atoms and molecules","volume":"27","author":"Svidzinsky","year":"2008","journal-title":"Int. Rev. Phys. Chem."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"090","DOI":"10.21468\/SciPostPhys.9.6.090","article-title":"Density scaling of generalized Lennard-Jones fluids in different dimensions","volume":"9","author":"Maimsbourg","year":"2020","journal-title":"SciPost Phys."},{"key":"ref_8","unstructured":"Dulieu, O., Colgan, J., Grant, E., Krishnakumar, E., Osterwalder, A., Sadeghpour, H., Vrakking, M., and Wu, J. (2018). Jubilee Issue of Hydrogen: A Fundamental System in All States, IOP Publishing. Special issue of Journal of Physics B."},{"key":"ref_9","unstructured":"Bharti, K., Ray, M., Varvitsiotis, A., Cabello, A., and Kwek, L.C. (2019). Local certification of programmable quantum devices of arbitrary high dimensionality. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"030308","DOI":"10.1103\/PRXQuantum.2.030308","article-title":"Infinite-dimensional programmable quantum processors","volume":"2","author":"Gschwendtner","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2909","DOI":"10.1007\/s10773-021-04889-8","article-title":"A note on quantum Bell nonlocality and quantum entanglement for high dimensional quantum systems","volume":"60","author":"Zhang","year":"2021","journal-title":"Int. J. Theoret. Phys."},{"key":"ref_12","unstructured":"Corda, C. (2020). On black hole Schr\u00f6dinger equation and gravitational fine structure constant. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Chen, X., Zhang, J.H., and Zhang, F.L. (2021). Probabilistic resumable quantum teleportation in high dimensions. arXiv.","DOI":"10.1088\/1674-1056\/ac1efb"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Anwar, A., Prabhakar, S., and Singh, R.P. (2021). Size invariant twisted optical modes for efficient generation of higher dimensional quantum states. arXiv.","DOI":"10.1364\/JOSAB.436088"},{"key":"ref_15","unstructured":"Achatz, L., Ortega, E., Dovzhik, K., Shiozaki, R.F., Fuenzalida, J., Wengerowsky, S., Bohmann, M., and Ursin, R. (2021). High-dimensional EPR entanglement from a SPDC source at telecom wavelength. arXiv."},{"key":"ref_16","unstructured":"Kopf, L., Hiekkamaki, M., Prabhakar, S., and Fickler, R. (2021). Endless fun in high dimensions-A quantum card game. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/0370-1573(90)90048-7","article-title":"Large-N expansions in quantum mechanics, atomic physics and some O(N) invariant systems","volume":"186","author":"Chatterjee","year":"1990","journal-title":"Phys. Rep."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.1088\/0305-4470\/24\/9\/022","article-title":"Shapes of random walks at order 1\/d2","volume":"24","author":"Beldjenna","year":"1991","journal-title":"J. Phys. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"6547","DOI":"10.1103\/PhysRevD.50.6547","article-title":"Scalar Casimir effect for a D-dimensional sphere","volume":"50","author":"Bender","year":"1994","journal-title":"Phys. Rev. D"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Kleftogiannos, I., and Amanatidis, I. (2021). Physics in non-fixed spatial dimensions. arXiv.","DOI":"10.1103\/PhysRevE.105.024141"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Tsipis, C.T., Popov, V.S., Herschbach, D.R., and Avery, J.S. (1996). New Methods in Quantum Theory, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-009-0227-5"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1002\/(SICI)1097-461X(1996)57:3<295::AID-QUA3>3.0.CO;2-T","article-title":"Dimensional scaling and renormalization","volume":"57","author":"Herschbach","year":"1996","journal-title":"Int. J. Quantum Chem."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"331","DOI":"10.3389\/fphy.2020.00331","article-title":"Unorthodox dimensional interpolations for He, Li, Be atoms and hydrogen molecule","volume":"8","author":"Ghosh","year":"2020","journal-title":"Front. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"103504","DOI":"10.1063\/1.2357998","article-title":"Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials","volume":"47","author":"Romera","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"065006","DOI":"10.1088\/1361-6455\/abcdee","article-title":"Multidimensional hydrogenic states: Position and momentum expectation values","volume":"54","author":"Dehesa","year":"2021","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"e26077","DOI":"10.1002\/qua.26077","article-title":"Analytical Shannon information entropies for all discrete multidimensional hydrogenic states","volume":"120","author":"Toranzo","year":"2020","journal-title":"Int J Quantum Chem."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"073203","DOI":"10.1088\/1742-5468\/aacf0c","article-title":"R\u00e9nyi entropies for multidimensional hydrogenic systems in position and momentum spaces","volume":"2018","author":"Toranzo","year":"2018","journal-title":"Stat. Mech. Theory Exp."},{"key":"ref_28","unstructured":"Olver, F.W.J., Lozier, D.W., Boisvert, R.F., and Clark, C.W. (2010). NIST Handbook of Mathematical Functions, Cambridge University Press."},{"key":"ref_29","unstructured":"Srivastava, H.M., and Karlsson, P.W. (1985). Multiple Gaussian Hypergeometric Series, Halsted Press-Ellis Horwood Limited."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/BF00641720","article-title":"A unified theory of polynomial expansions and their applications involving Clebsch-Gordan type linearization relations and Neumann series","volume":"150","author":"Srivastava","year":"1988","journal-title":"Astr. Sp. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.amc.2013.07.076","article-title":"R\u00e9nyi entropies, Lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions","volume":"223","author":"Dehesa","year":"2013","journal-title":"Appl. Math. Comp."},{"key":"ref_32","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_33","doi-asserted-by":"crossref","first-page":"103302","DOI":"10.1063\/1.5006569","article-title":"Entropic uncertainty measures for large dimensional hydrogenic systems","volume":"58","author":"Temme","year":"2017","journal-title":"J. Math. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"215206","DOI":"10.1088\/1751-8121\/aa6dc1","article-title":"Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters","volume":"50","author":"Temme","year":"2017","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Temme, N.M. (2015). Asymptotic Methods for Integrals, World Scientific.","DOI":"10.1142\/9195"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3065","DOI":"10.1103\/PhysRevA.50.3065","article-title":"Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom","volume":"50","author":"Dehesa","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/S0375-9601(01)00827-1","article-title":"Existence of bound states in continuous 0< D < \u221e dimensions","volume":"293","author":"Nieto","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1002\/qua.22244","article-title":"Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states","volume":"110","author":"Dehesa","year":"2010","journal-title":"Int. J. Quant. Chem."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Dong, S.H. (2011). Wave Equations in Higher Dimensions, Springer.","DOI":"10.1007\/978-94-007-1917-0"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1080\/01442350110075926","article-title":"Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem","volume":"20","author":"Aquilanti","year":"2001","journal-title":"Int. Rev. Phys. Chem."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Avery, J., and Avery, J. (2006). Generalized Sturmians and Atomic Spectra, World Sci. Publ.","DOI":"10.1142\/9789812773593"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/B978-0-12-411544-6.00005-4","article-title":"d-Dimensional Kepler-Coulomb Sturmians and hyperspherical harmonics as complete orthonormal atomic and molecular orbitals","volume":"67","author":"Coletti","year":"2013","journal-title":"Adv. Quantum Chem."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/BF01336904","article-title":"The hydrogen atom and non-Euclidean geometry","volume":"98","author":"Fock","year":"1935","journal-title":"Z. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1103\/PhysRev.34.109","article-title":"The momentum distribution in hydrogen-like atoms","volume":"34","author":"Podolsky","year":"1929","journal-title":"Phys. Rev."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1119\/1.17405","article-title":"On the momentum representation of hydrogenic wave functions: Some properties and an application","volume":"61","author":"Hey","year":"1993","journal-title":"Am. J. Phys."},{"key":"ref_46","unstructured":"Luke, Y.L. (1969). The Special Functions and Their Approximations, Academic Press."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1103\/PhysRevA.42.1123","article-title":"Expectation values of rP for arbitrary hydrogenic states","volume":"42","author":"Drake","year":"1990","journal-title":"Phys. Rev. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4435","DOI":"10.1088\/0953-4075\/30\/20\/008","article-title":"Recursive evaluation of expectation values for arbitrary states of the relativistic one-electron atom","volume":"30","author":"Andrae","year":"1997","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"3177","DOI":"10.1142\/S0217979204026408","article-title":"Exact numerical values of diagonal matrix elements < rk > nl, as n \u2264 8 and \u22127 \u2264 k \u2264 4, and the symmetry of Appell\u2019s function F2(1,1)","volume":"18","author":"Tarasov","year":"2004","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1134\/S0030400X15010233","article-title":"Radial matrix elements and the angular momentum technique","volume":"118","author":"Varshalovich","year":"2015","journal-title":"Opt. Spectrosc."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3884","DOI":"10.1137\/080736740","article-title":"A general asymptotic expansion formula for integrals involving high-order orthogonal polynomials","volume":"31","author":"Abrahams","year":"2009","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"6600","DOI":"10.1063\/1.1286984","article-title":"Functionals of Gegenbauer polynomials and D-dimensional hydrogenic momentum expectation values","volume":"41","author":"Dehesa","year":"2000","journal-title":"J. Math. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"082109","DOI":"10.1063\/1.4961322","article-title":"Heisenberg-like uncertainty measures for D-dimensional hydrogenic systems at large D","volume":"57","author":"Toranzo","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1007\/BF01397280","article-title":"\u00dcber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik","volume":"43","author":"Heisenberg","year":"1927","journal-title":"Z. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1007\/BF01391200","article-title":"Zur Quantenmechanik einfacher Bewegungstypen","volume":"44","author":"Kennard","year":"1927","journal-title":"Z. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1088\/1367-2630\/8\/12\/330","article-title":"Improvement of the Heisenberg and Fisher-information-based uncertainty relations for D-dimensional central potentials","volume":"8","author":"Dehesa","year":"2006","journal-title":"New J. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1103\/PhysRevA.50.311","article-title":"Information entropy and uncertainty in D-dimensional many-body systems","volume":"50","author":"Angulo","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"062102","DOI":"10.1103\/PhysRevA.83.062102","article-title":"Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms","volume":"83","author":"Angulo","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"042105","DOI":"10.1103\/PhysRevA.84.042105","article-title":"Upper bounds on quantum uncertainty products and complexity measures","volume":"84","author":"Guerrero","year":"2011","journal-title":"Phys. Rev."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"083102","DOI":"10.1088\/1742-5468\/aa7df4","article-title":"Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems","volume":"2017","author":"Toranzo","year":"2017","journal-title":"Stat. Mech. Theory Exp."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Puertas-Centeno, D., Toranzo, I.V., and Dehesa, J.S. (2017). Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems. Entropy, 19.","DOI":"10.3390\/e19040164"},{"key":"ref_62","first-page":"547","article-title":"On Measures of Entropy and Information","volume":"Volume 1","year":"1961","journal-title":"Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability"},{"key":"ref_63","unstructured":"Aczel, J., and Daroczy, Z. (1975). On Measures of Information and Their Characterizations, Academic Press."},{"key":"ref_64","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_65","doi-asserted-by":"crossref","first-page":"052101","DOI":"10.1103\/PhysRevA.74.052101","article-title":"Formulation of the uncertainty relations in terms of the R\u00e9nyi entropies","volume":"74","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"4800","DOI":"10.1016\/j.physa.2008.04.010","article-title":"Some extensions of the uncertainty principle","volume":"387","author":"Zozor","year":"2008","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_67","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\/23\/10\/1339\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:14:20Z","timestamp":1760166860000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/10\/1339"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,14]]},"references-count":67,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["e23101339"],"URL":"https:\/\/doi.org\/10.3390\/e23101339","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,10,14]]}}}