Abstract
In the Copenhagen interpretation the Heisenberg inequality ΔQΔP≥ℏ/2 is interpreted as the mathematical expression of the concept of complementarity, quantifying the mutual disturbance necessarily taking place in a simultaneous or joint measurement of incompatible observables. This interpretation was criticized a long time ago and has recently been challenged in an experimental way. These criticisms can be substantiated by using the generalized formalism of positive operator-valued measures, from which an inequality, different from the Heisenberg inequality, can be derived, precisely illustrating the Copenhagen concept of complementarity. The different roles of preparation and measurement in creating uncertainty in quantum mechanics are discussed.
Similar content being viewed by others
REFERENCES
W. Heisenberg, Z. Phys. 43, 172 (1927).
L. E. Ballentine, Rev. Mod. Phys. 42, 358 (1970).
G. Möllenstedt and C. Jönsson, Z. Phys 155, 472 (1959).
J. H. Shapiro and S. S. Wagner, J. Quant. Electr. QE20, 803 (1984).
N. G. Walker and J. E. Caroll, Electr. Lett. 20, 981 (1984).
H. Rauch, in Les fondements de la mécanique quantique, G. Gruber et al., eds. (25e Cours de perfectionnement de l'Association Vaudoise des Chercheurs en Physique, 1983), p. 330.
G. Badurek, H. Rauch, and D. Tuppinger, Phys. Rev. A 34, 2600 (1986).
E. Arthurs and J. L. Kelly Jr., Bell Syst. Techn. J. 44, 725 (1965).
Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, and G. Björk, in Progress in Optics XXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1990), p. 87.
S. Duerr, T. Nonn, and G. Rempe, Nature 395, 33 (1998).
E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, Nature 375, 368 (1995).
M. O. Scully, B.-G. Englert, and H. Walther, Nature 351, 111 (1991).
E. B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976).
A. S. Holevo, Probabilistic and Statistical Aspect of Quantum Theory (North-Holland, Amsterdam, 1982).
G. Ludwig, Foundations of Quantum Mechanics (Springer, Berlin, 1983), Vols. I and II.
P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Mechanics (Springer, Berlin, 1995).
H. Martens and W. De Muynck, Found. Phys. 20, 255, 357 (1990).
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
W. Heisenberg, The Physical Principles of Quantum Theory (Dover, New York, 1930).
P. Jordan, Erkenntnis 4, 215 (1934).
N. Bohr, in N. Bohr, Collected Works, J. Kalckar, ed. (North-Holland, Amsterdam, 1985), Vol. 6, p. 113.
N. Bohr, in Albert Einstein: Philosopher- Scientist, P. A. Schilpp, ed. (Open Court, La Salle, IL, 1949), p. 199.
E. Scheibe, The Logical Analysis of Quantum Mechanics (Pergamon Press, Oxford, 1973), p. 42.
E. Wigner, in Perspectives in Quantum Theory, W. Yourgrau and A. van der Merwe, eds. (MIT Press, Cambridge, MA, 1971), p. 25.
E. Arthurs and M. Goodman, Phys. Rev. Lett 60, 2447 (1988).
M. G. Raymer, Am. J. Phys. 62, 986 (1994).
H. Martens and W. De Muynck, J. Phys. A Math. Gen. 25, 4887 (1992).
D. M. Appleby, Int. J. Theor. Phys. 37, 1491 (1998).
H. M. Wiseman, Found. Phys. 28, 1619 (1998).
F. R. Gantmacher, Application of the Theory of Matrices (Interscience, New York, 1959).
P. L. Kelley and W. H. Kleiner, Phys. Rev. A 136, 316 (1964).
R. McEliece, The Theory of Information and Coding (Addison- Wesley, London, 1977).
W. M. de Muynck, Found. Phys. 14, 199 (1984).
W. M. de Muynck and H. Martens, Phys. Rev. A 42, 5079 (1990).
W. M. de Muynck, W. W. Stoffels, and H. Martens, Physica B 175, 127 (1991).
H. Martens and W. M. de Muynck, J. Phys. A Math. Gen. 26, 2001 (1993).
W. M. de Muynck and H. Martens, in Symmetry and Structural Properties of Condensed Matter, W. Florek, D. Lipinski, and T. Lulek, eds. (World Scientific, Singapore, 1993), p. 101.
S. V. Dorofeev and J. de Graaf, Indag. Mathem. N.S. 8, 349 (1997).
D. Deutsch, Phys. Rev. Lett. 50, 631 (1983).
M. H. Partovi, Phys. Rev. Lett. 50, 1883 (1983).
K. Kraus, Phys. Rev. D 35, 3070 (1987).
H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).
M. Brune, E. Hagley, J. Dreyer, X. Maî tre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 77, 4887 (1996).
Norman F. Ramsey, Molecular Beams (Clarendon Press, Oxford, 1956); reprinted lithographically in Great Brittain from corrected sheets of the first edition in 1963 and 1969.
J. Summhammer, H. Rauch, and D. Tuppinger, Phys. Rev. A 36, 4447 (1987).
B.-G. Englert, M. O. Scully, and H. Walther, Nature 375, 367 (1995).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
de Muynck, W.M. Preparation and Measurement: Two Independent Sources of Uncertainty in Quantum Mechanics. Foundations of Physics 30, 205–225 (2000). https://doi.org/10.1023/A:1003640932686
Issue Date:
DOI: https://doi.org/10.1023/A:1003640932686