×

What is the uncertainty principle of non-relativistic quantum mechanics? (English) Zbl 1392.81163

Summary: After more than ninety years of discussions over the uncertainty principle, there is still no universal agreement on what the principle states. The Robertson uncertainty relation (incorporating standard deviations) is given as the mathematical expression of the principle in most quantum mechanics textbooks. However, the uncertainty principle is not merely a statement of what any of the several uncertainty relations affirm. It is suggested that a better approach would be to present the uncertainty principle as a statement about the probability distributions of incompatible variables and the resulting restrictions on quantum states.

MSC:

81S05 Commutation relations and statistics as related to quantum mechanics (general)
62J10 Analysis of variance and covariance (ANOVA)
81P05 General and philosophical questions in quantum theory
Full Text: DOI

References:

[1] Busch P, Heinonen T and Lahti P 2007 Phys. Rep.452 155-76
[2] Bowman G E 2008 Essential Quantum Mechanics (Oxford: Oxford University Press) p 88 · Zbl 1201.81001
[3] Jijnasu V 2016 Stud. Hist. Phil. Mod. Phys.55 62-71 · Zbl 1348.81039
[4] Bricmont J 2016 Making Sense of Quantum Mechanics (Cham: Springer) p 58 · Zbl 1453.00008
[5] Hilgevoord J 1996 Am. J. Phys.64 1451-6
[6] Home D 1997 Conceptual Foundations of Quantum Physics: An Overview from Modern Perspectives (New York and London: Plenum) p 67 · Zbl 0972.81001
[7] Ozawa M 2003 Phys. Lett. A 318 21-9 · Zbl 1098.81747
[8] Werner R F 2004 Quantum Inf. Comput.4 547
[9] Faddeev L D and Yakubovskiĭ O A 2009 Lectures on Quantum Mechanics for Mathematics Students (Providence, RI: American Mathematical Society) p 21 · Zbl 1193.81001
[10] Zettili N 2009 Quantum Mechanics: Concepts and Applications (Chichester: Wiley) p 172
[11] Heisenberg W 1927 Z. Phys.43 172-98 · JFM 53.0853.05
[12] Reprinted in English inWheeler J A and Zurek W H (ed) 1983 Quantum Theory and Measurement (Princeton, NJ: Princeton University Press) pp 64-5
[13] Heisenberg W 1983 Quantum Theory and Measurement ed J A Wheeler and W H Zurek (Princeton, NJ: Princeton University Press) p 83
[14] Cramer J G 2016 The Quantum Handshake: Entanglement, Nonlocality and Transactions (Cham: Springer) pp 22-3 · Zbl 1358.81001
[15] Brown H R and Redhead M L G 1981 Found. Phys.11 1-20
[16] Wallace-Garden R 1984 Modern Logic and Quantum Mechanics (Bristol: Adam Hilger) pp 15-6 · Zbl 0586.03003
[17] Redhead M 1987 Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics (Oxford: Clarendon) pp 67-9 · Zbl 0611.03003
[18] Rozema L A et al 2015 Quantum Stud.: Math. Found.2 17-22 · Zbl 1317.81015
[19] Kennard E H 1927 Z. Phys.44 326-52 · JFM 53.0853.02
[20] Erhart J et al 2012 Nat. Phys.8 185-9
[21] Rozema L A et al 2012 Phys. Rev. Lett.109 100404
[22] Sulyok G et al 2013 Phys. Rev. A 88 022110
[23] Ringbauer M et al 2014 Phys. Rev. Lett.112 020401
[24] Kaneda F et al 2014 Phys. Rev. Lett.112 020402
[25] Busch P et al 2014 Rev. Mod. Phys.86 1261
[26] Ballentine L E 1998 Quantum Mechanics: A Modern Development (Singapore: World Scientific) pp 45-6 · Zbl 0997.81501
[27] Home D 1997 Conceptual Foundations of Quantum Physics: An Overview from Modern Perspectives (New York and London: Plenum) p 3 · Zbl 0972.81001
[28] Penrose R 2004 The Road to Reality: A Complete Guide to the Laws of the Universe (London: Jonathon Cape) p 517 · Zbl 1188.00007
[29] Wallace-Garden R 1984 Modern Logic and Quantum Mechanics (Bristol: Adam Hilger) p 11 · Zbl 0586.03003
[30] Audretsch J 2007 Entangled Systems: New Directions in Quantum Physics (Weinheim: Wiley-VCH) p 19 · Zbl 1136.81001
[31] Weyl H 1928 Gruppentheorie Und Quantenmechanik (Leipzig: Hirzel) · JFM 54.0954.03
[32] Schiff L I 1968 Quantum Mechanics (Tokyo: McGraw-Hill Kogakusha) p 7
[33] Fayer M D 2001 Elements of Quantum Mechanics (Oxford; New York: Oxford University Press) p 35
[34] Steward E G 2012 Quantum Mechanics: Its Early Development and the Road to Entanglement and Beyond (London: Imperial College Press) pp 131-2 · Zbl 1250.81007
[35] Hilgevoord J and Uffink J B M 2016 The uncertainty principle The Stanford Encyclopedia of Philosophy Fall 2016 edn ed E N Zalta (http://plato.stanford.edu/archives/fall2016/entries/qt-uncertainty/) · Zbl 0959.81502
[36] Robertson H P 1929 Phys. Rev.34 163-4
[37] Home D 1997 Conceptual Foundations of Quantum Physics: An Overview from Modern Perspectives (New York and London: Plenum) p 13 · Zbl 0972.81001
[38] Ballentine L E 1998 Quantum Mechanics: A Modern Development (Singapore: World Scientific) p 226 · Zbl 0997.81501
[39] Redhead M 1987 Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics (Oxford: Clarendon) p 62 · Zbl 0611.03003
[40] Redhead M 1987 Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics (Oxford: Clarendon) p 61 · Zbl 0611.03003
[41] Schrödinger E 1930 Berliner Berichte1930 296-303 · JFM 56.0754.05
[42] Reprinted in English inAngelow A and Batoni M C 1999 Bulg. J. Physics26 193-203
[43] Jammer M 1974 The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective (New York: Wiley) p 73
[44] Fayer M D 2001 Elements of Quantum Mechanics (Oxford; New York: Oxford University Press) p 43
[45] Sakurai J J 1985 Modern Quantum Mechanics (Redwood City, CA: Addison-Wesley) p 36
[46] Butterfield J 2013 On time in quantum mechanics A Companion to the Philosophy of Time ed H Dyke and A Bardon (Chichester: Wiley-Blackwell) pp 229-30
[47] Hilgevoord J and Uffink J B M 1985 Eur. J. Phys.6 165-70
[48] Uffink J B M and Hilgevoord J 1985 Found. Phys.15 925-44
[49] Hilgevoord J 2002 Am. J. Phys.70 983
[50] Hilgevoord J and Uffink J B M 1988 The mathematical expression of the uncertainty principle Microphysical Reality and Quantum Formalism vol 1 ed A van der Merwe et al (Dordrecht: Reidel Academic) pp 96-101
[51] Landsberg T 1990 Uncertainty and measurement Quantum Theory Without Reduction ed M Cini and J-M Lévy-Leblond (Bristol: Adam Hilger)
[52] Landau H J and Pollak H O 1961 Bell Syst. Tech. J.40 65-84 · Zbl 0184.08602
[53] Hilgevoord J and Uffink J B M 1990 A new look on the uncertainty principle Sixty-two Years of Uncertainty: Historical, Philosophical and Physical Inquiries into the Foundations of Quantum Mechanics ed A I Miller (New York: Plenum) p 128
[54] Weston M M et al 2013 Phys. Rev. Lett.110 220402
[55] Baek S Y et al 2013 Sci. Rep.3 2221 (http://nature.com/articles/srep02221)
[56] Dressel J and Nori F 2014 Phys. Rev. A 89 022106
[57] Busch P, Lahti P and Werner R F 2014 Rev. Mod. Phys.86 1261-81
[58] Dick R 2016 Advanced Quantum Mechanics: Materials and Photons (Cham: Springer) p 86 · Zbl 1375.81005
[59] Bongaarts P 2015 Quantum Theory: A Mathematical Approach (Cham: Springer) p 60 · Zbl 1315.81001
[60] Pade J 2014 Quantum Mechanics for Pedestrians 1: Fundamentals (Cham: Springer) p 172 · Zbl 1278.81003
[61] Weinberg S 2013 Lectures on Quantum Mechanics (Cambridge: Cambridge University Press) pp 65-6 · Zbl 1264.81010
[62] McIntyre D H 2012 Quantum Mechanics: A Paradigms Approach (Boston, MA: Pearson) pp 56, 63
[63] Razavy M 2011 Heisenberg’s Quantum Mechanics (Singapore: World Scientific) pp 91-5 · Zbl 1230.81001
[64] Jaeger G 2014 Quantum Objects: Non-Local Correlations, Causality and Objective Indefiniteness in the Quantum World (Berlin: Springer) ch 2
[65] Jammer M 1974 The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective (New York: Wiley) p 79
[66] SeeJammer M 1974 The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective (New York: Wiley) pp 9-17 for further discussion
[67] Ho-Kim Q, Kumar N and Lam C S 2004 Invitation to Contemporary Physics (Hackensack, NJ: World Scientific) p 467 · Zbl 1060.81002
[68] Greenstein G and Zajonc A G 2006 The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics (Sudbury, MA: Jones and Bartlett Publishers) p 53
[69] Cushing J T 1998 Philosophical Concepts in Physics: The Historical Relation Between Philosophy and Scientific Theories (Cambridge: Cambridge University Press) p 301 · Zbl 0951.01010
[70] Dickson M 2007 Non-relativistic quantum mechanics Philosophy of Physics ed J Butterfield and J Earman (Amsterdam: Elsevier) p 348
[71] Sudbery A 1986 Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians (Cambridge: Cambridge University Press) pp 210-25 · Zbl 0622.46052
[72] Dickson M 2007 Non-relativistic quantum mechanics Philosophy of Physics ed J Butterfield and J Earman (Amsterdam: Elsevier) pp 368-73
[73] Pykacz J 2015 A brief survey of main interpretations of quantum mechanics Quantum Physics, Fuzzy Sets and Logic ed J Pykacz (Cham; New York: Springer) · Zbl 1328.81020
[74] Heisenberg W 1958 Physics and Philosophy: The Revolution in Modern Science (London: Penguin) ch 3
[75] Jammer M 1966 The Conceptual Development of Quantum Mechanics (New York: McGraw-Hill) p 361
[76] Baggott J 1992 The Meaning of Quantum Theory (Oxford: Oxford University Press) p 82
[77] Cushing J T 1994 Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony (Chicago, IL: University of Chicago Press) p 289 · Zbl 0828.00005
[78] Faye J 2014 Copenhagen interpretation of quantum mechanics The Stanford Encyclopedia of Philosophy Fall 2014 edn ed E N Zalta (http://plato.stanford.edu/archives/fall2014/entries/qm-copenhagen/)
[79] Steward E G 2012 Quantum Mechanics: Its Early Development and the Road to Entanglement and Beyond (London: Imperial College Press) p 144 · Zbl 1250.81007
[80] de la Peña L, Cetto A M and Hernández A V 2015 The Emerging Quantum: The Physics Behind Quantum Mechanics (Cham; New York: Springer) p 14 · Zbl 1300.81001
[81] Wallace-Garden R 1984 Modern Logic and Quantum Mechanics (Bristol: Adam Hilger) p 14 · Zbl 0586.03003
[82] Griffiths D J 2005 Introduction to Quantum Mechanics (Upper Saddle River, NJ: Pearson Prentice-Hall) p 112
[83] Sen D 2014 Curr. Sci.107 203-18 (http://www.currentscience.ac.in/cs/Volumes/107/02/0203.pdf)
[84] Bowman G E 2008 Essential Quantum Mechanics (Oxford: Oxford University Press) p 86 · Zbl 1201.81001
[85] Müller R and Wiesner H 2002 Am. J. Phys.70 200-9
[86] Narasimhachar V, Poostindouz A and Gour G 2015 The principle behind the uncertainty principle arXiv:1505.02223v1
[87] Singh C and Marshman E 2015 Phys. Rev. Spec. Top.—Phys. Educ. Res.11 020117
[88] Singh C 2008 Am. J. Phys.76 400
[89] Kohnle A et al 2014 Eur. J. Phys.35 015001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.