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A Note on Quantum Theory, Complementarity, and Uncertainty

Published online by Cambridge University Press:  01 April 2022

Paul Busch and
Pekka J. Lahti
Paul Busch
Affiliation:
Institute for Theoretical Physics, University of Cologne
Pekka J. Lahti
Affiliation:
Department of Physical Sciences, University of Turku
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Abstract

Uncertainty relations and complementarity of canonically conjugate position and momentum observables in quantum theory are discussed with respect to some general coupling properties of a function and its Fourier transform. The question of joint localization of a particle on bounded position and momentum value sets and the relevance of this question to the interpretation of position-momentum uncertainty relations is surveyed. In particular, it is argued that the Heisenberg interpretation of the uncertainty relations can consistently be carried through in a natural extension of the usual Hilbert space frame of the quantum theory.

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Type
Research Article
Information
Philosophy of Science , Volume 52 , Issue 1 , March 1985, pp. 64 - 77
Copyright
Copyright © 1985 by the Philosophy of Science Association

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References

Amrein, W., and Berthier, A. (1977), “On Support Properties of Lp-functions and Their Fourier Transformations”, Journal of Functional Analysis 24: 258–67.CrossRefGoogle Scholar
Ballentine, L. E. (1970), “The Statistical Interpretation of Quantum Mechanics”, Review of Modern Physics 42: 358–81.CrossRefGoogle Scholar
Berthier, A., and Jauch, J. M. (1976), “Theorem on the support of functions in L 2(R) and of their Fourier transform”, Letters in Mathematical Physics 1: 93–97.CrossRefGoogle Scholar
Bohr, N. (1928), “The Quantum Postulate and the Recent Development of Atomic Theory”, Nature 121: 580–90.CrossRefGoogle Scholar
Bohr, N. (1949), “Discussion with Einstein on Epistemological Problems in Atomic Physics”, in Albert Einstein: Philosopher-Scientist. Evanston: The Library of Living Philosophers, Inc.Google Scholar
Busch, P. (1982), “Unbestimmtheitsrelation und simultane Messungen in der Quantentheorie”, Ph.D. dissertation, University of Cologne.Google Scholar
Busch, P. (1984), “On joint lower bounds of position and momentum observables in quantum mechanics”, Journal of Mathematical Physics 25: 1794–97.CrossRefGoogle Scholar
Busch, P. (1985), “Indeterminacy Relations and Simultaneous Measurements in Quantum Theory”, International Journal of Theoretical Physics 24: 63–92.CrossRefGoogle Scholar
Busch, P., and Lahti, P. (1984), “On various joint measurements of position and momentum observables”, Physical Review D 29: 1634–46.CrossRefGoogle Scholar
Davies, E. B. (1976), Quantum Theory and Open Systems. London: Academic Press.Google Scholar
Drieschner, M. (1979), Voraussage, Wahrscheinlichkeit, Object. Springer Lecture Notes in Physics 99. Berlin: Springer-Verlag.Google Scholar
Drieschner, M. (1981), Einführung in die Naturphilosophie. Darmstadt: Wissenschaftliche Buchgesellschaft.Google Scholar
Einstein, A. (1949), Albert Einstein: Philosopher-Scientist. Evanston: The Library of Living Philosophers, Inc.Google Scholar
Fock, V. (1978), Fundamentals of Quantum Mechanics. Moscow: Mir Publishers.Google Scholar
Gibbins, P. (1981), “A Note on Quantum Logic and the Uncertainty Principle”, Philosophy of Science 48: 122–26.CrossRefGoogle Scholar
Heisenberg, W. (1927), “Über den anschaulichen Inhalt der quanten-theoretischen Kinematik und Mechanik”, Zeitschrift für Physik 43: 172–98.CrossRefGoogle Scholar
Heisenberg, W. (1949), The Physical Principles of the Quantum Theory. New York: Dover. (First published by The University of Chicago Press, 1930.)Google Scholar
Heisenberg, W. (1958), Physics and Philosophy. New York: Harper & Row.Google Scholar
Jammer, M. (1979), “A consideration of the philosophical implications of the new physics”, in The Structure and Development of Science, Radnitzky, G. and Andersson, G. (eds.). Dordrecht: D. Reidel.Google Scholar
Jammer, M. (1981), “Zu den philosophischen Konsequenzen der neuen Physik”, in Voraussetzungen und Grenzen der Wissenschaft, Radnitzky, G. and Andersson, G. (eds.). Tübingen: Mohr.Google Scholar
Jammer, M. (1982), “A Note on Peter Gibbins' ‘A Note on Quantum Logic and the Uncertainty Principle‘”, Philosophy of Science 49: 478–79.CrossRefGoogle Scholar
Jauch, J. M. (1968), Foundations of Quantum Mechanics. Reading, Mass.: Addison-Wesley Publishing Company.Google Scholar
Jauch, J. M. (1976), “The Quantum Probability Calculus”, in Logic and Probability in Quantum Mechanics, Suppes, P. (ed.). Dordrecht: D. Reidel. (First published in Synthese 29: 131–54.)Google Scholar
Lahti, P. J. (1979), “Uncertainty and Complementarity in Axiomatic Quantum Mechanics”. Report Series No. D2, Theoretical Physics, Department of Physical Sciences, University of Turku, Finland. Reprinted in International Journal of Theoretical Physics 19: 789–842.CrossRefGoogle Scholar
Lahti, P. J. (1980), “Characterization of Quantum Logics”, International Journal of Theoretical Physics 19: 905–23.CrossRefGoogle Scholar
Lahti, P. J. (1983a), “Hilbertian quantum theory as the theory of complementarity”, International Journal of Theoretical Physics 22: 911–29.CrossRefGoogle Scholar
Lahti, P. J. (1983b), “Probability, uncertainty, and complementarity”, Report Series No. R43, Department of Physical Sciences, University of Turku, Finland.Google Scholar
Lenard, A. (1972), “The numerical range of a pair of projections”, Journal of Functional Analysis 10: 410–23.CrossRefGoogle Scholar
Ludwig, G. (1954), Die Grundlagen der Quantenmechanik. Berlin: Springer-Verlag, pp. 60–61.Google Scholar
Margenau, H. (1963), “Measurements and Quantum States”, Philosophy of Science 30: 1–16.CrossRefGoogle Scholar
Mehlberg, H. (1967), “The Problem of Physical Reality in Contemporary Science”, in Quantum Theory and Reality, Bunge, M. (ed.). Berlin: Springer-Verlag, pp. 45–65.CrossRefGoogle Scholar
Mittelstaedt, P. (1980), “The Concepts of Truth, Possibility, and Probability in the Language of Quantum Physics”, in Interpretations and Foundations of Quantum Theory, Neumann, A. (ed.). Mannheim: B. I. Wissenschaftsverlag.Google Scholar
Paley, R., and Wiener, N. (1934), Fourier Transforms in the Complex Domain. American Mathematical Society Colloquium Publications, vol. 19. New York.Google Scholar
Pauli, W. (1980), General Principles of Quantum Mechanics. Translated from the 1958 edition of Die allgemeinen Prinzipen der Wellenmechanik, 1933. Berlin: Springer-Verlag.Google Scholar
Piron, C. (1964), “Axiomatique Quantique”, Helvetica Physica Acta 37: 439–68.Google Scholar
Popper, K. R. (1957), “The propensity interpretation of the calculus of probability, and the quantum theory”, in Observation and Interpretation in the Philosophy of Physics, Körner, S. (ed.). London: Butterworth's Scientific Publications.Google Scholar
Popper, K. R. (1980), The Logic of Scientific Discovery. London: Hutchison. (First published in 1934.)Google Scholar
Primas, H. (1981), Chemistry, Quantum Mechanics, and Reductionism. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Prugovecki, E. (1981), Quantum Mechanics in Hilbert Space. 2nd revised edition. New York: Academic Press.Google Scholar
Reed, H., and Simon, B. (1975), Methods of Modern Mathematical Physics II: Fourier Analysis and Self-Adjointness. New York: Academic Press.Google Scholar
Reiter, H., and Thirring, W. (1983), “Are x and p incompatible observables?” Institut für Theoretische Physik, University of Vienna.Google Scholar
Scheibe, E. (1973), The Logical Analysis of Quantum Mechanics. Oxford: Pergamon Press.Google Scholar
Stachow, E.-W. (1981), “Der quantenlogische Wahrscheinlichkeitskalkül”, in Grundlagenprobleme der modernen Physik, Nitsch, J., Pfarr, J., and Stachow, E.-W. (eds.). Mannheim: B. I. Wissenschafstverlag.Google Scholar
Suppes, P. (1961), “Probability concepts in quantum mechanics”, Philosophy of Science 22: 378–89.Google Scholar
Titchmarsh, E. (1937), Introduction to the theory of Fourier integrals. Oxford: Clarendon Press.Google Scholar
von Weizsäcker, C. F. (1955), “Komplementarität und Logik”, Die Naturwissenschaften 42, No. 19/20.Google Scholar
Wigner, E. P. (1971), “Quantum-Mechanical Distribution Functions Revisited”, in Perspective in Quantum Physics: Essays in Honor of Alfred Lande, Yourgrau, W. and van der Merwe, A. (eds.). Cambridge and London: The MIT Press, pp. 25–36.Google Scholar