{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T02:38:00Z","timestamp":1792723080979,"version":"4.5.0"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,14]],"date-time":"2024-01-14T00:00:00Z","timestamp":1705190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2024,1,14]],"date-time":"2024-01-14T00:00:00Z","timestamp":1705190400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Cosmogenics Inc."}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The time\u2013energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the \u21131 norm of energy coherence C is invariant with respect to the unitary evolution of the quantum state. Thus, the \u21131 norm of energy coherence C of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal \u21131 norm of energy coherence, C&lt;1, the Mandelstam\u2013Tamm and Margolus\u2013Levitin relations generate instances of infinite \u201ctime uncertainty\u201d that is devoid of physical meaning. Only for quantum states with the maximal \u21131 norm of energy coherence, C=1, the Mandelstam\u2013Tamm and Margolus\u2013Levitin relations avoid infinite \u201ctime uncertainty\u201d, but they both reduce to a strict equality that expresses the Einstein\u2013Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schr\u00f6dinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.<\/jats:p>","DOI":"10.3390\/sym16010100","type":"journal-article","created":{"date-parts":[[2024,1,15]],"date-time":"2024-01-15T04:03:26Z","timestamp":1705291406000},"page":"100","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Time\u2013Energy Uncertainty Relation in Nonrelativistic Quantum Mechanics"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6846-1194","authenticated-orcid":false,"given":"Danko D.","family":"Georgiev","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,14]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","unstructured":"Wheeler, J.A., and Zurek, W.H. (1983). The physical content of quantum kinematics and mechanics. Quantum Theory and Measurement, Princeton University Press.","DOI":"10.1515\/9781400854554"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1038\/121580a0","article-title":"The quantum postulate and the recent development of atomic theory","volume":"121","author":"Bohr","year":"1928","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1103\/PhysRev.37.780","article-title":"Knowledge of past and future in quantum mechanics","volume":"37","author":"Einstein","year":"1931","journal-title":"Phys. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Born, M., and Jordan, P. (1930). Elementare Quantenmechanik: Zweiter Band der Vorlesungen \u00fcber Atommechanik, Springer.","DOI":"10.1007\/978-3-662-00291-9"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wheeler, J.A., and Zurek, W.H. (1983). Discussion with Einstein on epistemological problems in atomic physics. Quantum Theory and Measurement, Princeton University Press.","DOI":"10.1515\/9781400854554"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/S1876-0503(08)70379-7","article-title":"Discussion with Einstein on epistemological problems in atomic physics","volume":"Volume 7","author":"Kalckar","year":"1996","journal-title":"Niels Bohr Collected Works"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1103\/PhysRev.28.1049","article-title":"An undulatory theory of the mechanics of atoms and molecules","volume":"28","year":"1926","journal-title":"Phys. Rev."},{"key":"ref_9","unstructured":"Schr\u00f6dinger, E. (1928). Collected Papers on Wave Mechanics, Blackie & Son."},{"key":"ref_10","unstructured":"Dirac, P.A.M. (1967). The Principles of Quantum Mechanics, Oxford University Press. [4th ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bethe, H., Hund, F., Mott, N.F., Pauli, W., Rubinowicz, A., Wentzel, G., and Smekal, A. (1933). Die allgemeinen Prinzipien der Wellenmechanik. Quantentheorie, Springer. Handbuch der Physik.","DOI":"10.1007\/978-3-642-52619-0"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Pauli, W. (1980). General Principles of Quantum Mechanics, Springer.","DOI":"10.1007\/978-3-642-61840-6"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/BF01329203","article-title":"Bemerkung \u00fcber die angen\u00e4herte G\u00fcltigkeit der klassischen Mechanik innerhalb der Quantenmechanik","volume":"45","author":"Ehrenfest","year":"1927","journal-title":"Z. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1103\/PhysRev.34.163","article-title":"The uncertainty principle","volume":"34","author":"Robertson","year":"1929","journal-title":"Phys. Rev."},{"key":"ref_15","first-page":"296","article-title":"Zum Heisenbergschen Unsch\u00e4rfeprinzip","volume":"14","year":"1930","journal-title":"Sitzungsberichte Preuss. Akad. Wiss. Phys. Math. Kl."},{"key":"ref_16","first-page":"69","article-title":"The time\u2013energy uncertainty relation","volume":"Volume 72","author":"Muga","year":"2002","journal-title":"Time in Quantum Mechanics"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"453001","DOI":"10.1088\/1751-8121\/aa86c6","article-title":"Quantum speed limits: From Heisenberg\u2019s uncertainty principle to optimal quantum control","volume":"50","author":"Deffner","year":"2017","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_18","first-page":"249","article-title":"The uncertainty relation between energy and time in non-relativistic quantum mechanics","volume":"9","author":"Mandelstam","year":"1945","journal-title":"J. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/S0167-2789(98)00054-2","article-title":"The maximum speed of dynamical evolution","volume":"120","author":"Margolus","year":"1998","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"190","DOI":"10.12743\/quanta.v12i1.250","article-title":"The enigmas of fluctuations of the universal quantum fields","volume":"12","author":"Bhaumik","year":"2023","journal-title":"Quanta"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/0375-9601(92)90555-Z","article-title":"An energy dispersion estimate","volume":"161","author":"Uhlmann","year":"1992","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"055004","DOI":"10.1088\/1367-2630\/ac688a","article-title":"Extensions of the Mandelstam\u2013Tamm quantum speed limit to systems in mixed states","volume":"24","author":"Allan","year":"2022","journal-title":"New J. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bagchi, S., Thakuria, D., and Pati, A.K. (2023). Stronger quantum speed limit for mixed quantum states. Entropy, 25.","DOI":"10.20944\/preprints202305.1616.v1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"140401","DOI":"10.1103\/PhysRevLett.113.140401","article-title":"Quantifying coherence","volume":"113","author":"Baumgratz","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"24","DOI":"10.12743\/quanta.v8i1.87","article-title":"Coherence, interference and visibility","volume":"8","author":"Qureshi","year":"2019","journal-title":"Quanta"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1364\/OL.415556","article-title":"Predictability, distinguishability and entanglement","volume":"46","author":"Qureshi","year":"2021","journal-title":"Opt. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Peled, B.Y., Te\u2019eni, A., Georgiev, D., Cohen, E., and Carmi, A. (2020). Double slit with an Einstein\u2013Podolsky\u2013Rosen pair. Appl. Sci., 10.","DOI":"10.3390\/app10030792"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Georgiev, D.D. (2021). Quantum information in neural systems. Symmetry, 13.","DOI":"10.3390\/sym13050773"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Griffiths, D.J., and Schroeter, D.F. (2018). Introduction to Quantum Mechanics, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/9781316995433"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1112\/plms\/s1-29.1.14","article-title":"On a law of combination of operators (second paper)","volume":"29","author":"Campbell","year":"1897","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1112\/plms\/s2-3.1.24","article-title":"Alternants and continuous groups","volume":"2","author":"Baker","year":"1905","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_32","first-page":"19","article-title":"Die symbolische Exponentialformel in der Gruppentheorie","volume":"58","author":"Hausdorff","year":"1906","journal-title":"Berichte \u00dcber Die Verhandlungen K\u00f6niglich S\u00e4chsischen Ges. Wiss. Leipz. Math. Phys. Kl."},{"key":"ref_33","first-page":"323","article-title":"Calculation of the coefficients in the Campbell\u2013Hausdorff formula","volume":"57","author":"Dynkin","year":"1947","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"052109","DOI":"10.1103\/PhysRevA.67.052109","article-title":"Quantum limits to dynamical evolution","volume":"67","author":"Giovannetti","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"160502","DOI":"10.1103\/PhysRevLett.103.160502","article-title":"Fundamental limit on the rate of quantum dynamics: The unified bound is tight","volume":"103","author":"Levitin","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2250101","DOI":"10.1142\/S0217984922501019","article-title":"Sensitivity of entanglement measures in bipartite pure quantum states","volume":"36","author":"Georgiev","year":"2022","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.physa.2018.11.026","article-title":"On the quantum dynamics of Davydov solitons in protein \u03b1-helices","volume":"517","author":"Georgiev","year":"2019","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1063\/1.1696139","article-title":"On derivations of the uncertainty principle","volume":"42","author":"Davidson","year":"1965","journal-title":"J. Chem. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1007\/BF02838760","article-title":"Schr\u00f6dinger\u2019s uncertainty principle?","volume":"4","author":"Nityananda","year":"1999","journal-title":"Resonance"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1557","DOI":"10.1023\/B:JOSS.0000013971.75667.c8","article-title":"An informational characterization of Schr\u00f6dinger\u2019s uncertainty relations","volume":"114","author":"Luo","year":"2004","journal-title":"J. Stat. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/100\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T01:51:18Z","timestamp":1792720278000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/1\/100"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,14]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["sym16010100"],"URL":"https:\/\/doi.org\/10.3390\/sym16010100","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,14]]}}}