Abstract
The problem of simultaneous measurement of incompatible observables in quantum mechanics is studied on the one hand from the viewpoint of an axiomatic treatment of quantum mechanics and on the other hand starting from a theory of measurement. It is argued that it is precisely such a theory of measurement that should provide a meaning to the axiomatically introduced concepts, especially to the concept of observable. Defining an observable as a class of measurement procedures yielding a certain prescribed result for the probability distribution of the set of values of some quantity (to be described by the set of eigenvalues of some Hermitian operator), this notion is extended to joint probability distributions of incompatible observables. It is shown that such an extension is possible on the basis of a theory of measurement, under the proviso that in simultaneously measuring such observables there is a disturbance of the measurement results of the one observable, caused by the presence of the measuring instrument of the other observable. This has as a consequence that the joint probability distribution cannot obey the marginal distribution laws usually imposed. This result is of great importance in exposing quantum mechanics as an axiomatized theory, since overlooking it seems to prohibit an axiomatic description of simultaneous measurement of incompatible observables by quantum mechanics.
Similar content being viewed by others
References
J. von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton, 1955).
N. Bohr, inAlbert Einstein: Philosopher-Scientist, P. Schilpp, ed. (Tudor, 1949), pp. 201–241.
W. Heisenberg,The Physical Principles of Quantum Theory (Chicago, 1930);Z. Physik 43, 172 (1927).
J. L. Park and H. Margenau,Int. J. Theor. Phys. 1, 211 (1968).
J. L. Park,Phil. of Science 35, 205, 389 (1968).
L. E. Ballentine,Rev. Mod. Phys. 42, 358 (1970).
L. S. Mayants, W. Yourgrau, and A. J. van der Merwe,Ann. d. Physik 7. Folge 33, 21 (1976).
J. M. Jauch,Foundations of Quantum Mechanics (Addison-Wesley, Reading, Mass., 1968).
J. Bub,The Interpretation of Quantum Mechanics (Reidel, Dordrecht, 1974).
J. M. Jauch,Synthese 29, 131 (1974).
D. I. Blokhintsev,Sov. Phys.—Uspekhi 11, 320 (1968); alsoThe Philosophy of Quantum Mechanics (Reidel, Dordrecht, 1968).
F. J. Belinfante,Measurements and Time Reversal in Objective Quantum Theory (Pergamon Press, 1975).
A. Daneri, A. Loinger, and G. M. Prosperi,Nucl. Phys. 33, 297 (1962); L. Rosenfeld,Progr. Theor. Phys. Suppl. (Commemoration Issue for the 30th Anniversary of the Meson Theory by Dr. H. Yukawa)1965, 222.
P. A. Moldauer,Phys. Review D 5, 1028 (1972).
H. Margenau,The Nature of Physical Reality (McGraw-Hill, 1950), Chapter 18.
E. Wigner,Phys. Rev. 40, 749 (1932).
L. Cohen,J. Math. Phys. 7, 781 (1966); G. S. Agarwal and E. Wolf,Phys. Rev. D 2, 2161, 2187, 2206 (1970).
L. Cohen,Phil. of Science 33, 317 (1966).
E. P. Wigner, inPerspectives in Quantum Theory, W. Yourgrau and A. van der Merwe, eds. (MIT Press, Cambridge, Mass., 1971), pp. 25–36.
M. D. Srinivas and E. Wolf,Phys. Rev. D 11, 1477 (1975).
P. R. Halmos,A Hilbert Space Problem Book (Van Nostrand, New York, 1967), p. 58.
D. J. Ross,Synthese 29, 373 (1974).
V. S. Varadarajan,Comm. Pure Appl. Math. XV, 189 (1962).
J. S. Bell,Rev. Mod. Phys. 38, 447 (1966); J. S. Bell, inRendiconti della Scuola Internazionale di Fisica -Enrico Fermi-, IL Corso (Academic Press, New York, 1971), p. 171.
L. S. Mayants,Found. Phys. 3, 413 (1973).
H. Margenau and R. N. Hill,Progr. Theor. Phys. 26, 722 (1961); E. P. Wigner,Am. J. Phys. 38, 1005 (1970).
L. S. Mayants,Found. Phys. 4, 335 (1974).
S. Kochen and E. P. Specker,J. Math. Mech. 17, 59 (1967).
H. J. Groenewold,Physica 12, 405 (1946).
N. G. de Bruijn,Nieuw Archief voor Wiskunde (3)XXI, 205 (1973); N. D. Cartwright,Physica 83A, 210 (1976).
G. Helmberg,Introduction to Spectral Theory in Hilbert Space (North-Holland, Amsterdam, 1969).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
de Muynck, W.M., Janssen, P.A.E.M. & Santman, A. Simultaneous measurement and joint probability distributions in quantum mechanics. Found Phys 9, 71–122 (1979). https://doi.org/10.1007/BF00715052
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00715052