# The Problem of Measurement

Chapter

## Abstract

The standard theory of measurements in quantum mechanics is reviewed with special emphasis on the conceptual and epistemological implications. It is concluded that the standard theory remains the only one which is compatible with present quantum mechanics. Hence, if one wants to avoid the conclusion that quantum mechanics only gives probability connections between subsequent observations, the quantum-mechanical equations would have to be modified. Particular attention is paid to the case that the measuring apparatus is macroscopic and its state vector not accurately known before the measurement.

## Preview

Unable to display preview. Download preview PDF.

## Reference

- 1.Some of the more recent papers on the subject are: Y. Aharonov and D.Bohm, Phys. Rev. 122, 1649 (1961); Nuovo cimento 17, 964 (1960); B. Bertotti, Nuovo cimento Suppl. 17, 1 (1960); L. de Broglie, J. phys. radium 20, 963 (1959); J. A. de Silva, Ann. In st. Henri Poincaré 16, 289 (1960); A. Datzeff, Compt. rend. 251, 1462 (1960); J. phys. radium 21, 201 (1960); 22, 101 (1961); J. M. Jauch, Hely. Phvs. Acta 33, 711 (1960); A. Landé, Z. Physik 162, 410 (1961); 164, 558 (1961); Am. J. Phvs. 29, 503 (1961); H. Margenau and R. N. Hill, Progr. Theo¬ret. Phys. 26, 727 (1961); A. Peres and P. Singer, Nuovo cimento 15, 907 (1960); H. Putnam, Phil. Sci. 28, 234 (1961); M. Renninger, Z. Physik 158, 417 (1960); L. Rosenfeld, Nature 190, 384 (1961); F. Schlögl, Z. Physik 159, 411 (1960); J. Schwinger, Proc. Natl. Acad. Sci. U. S. 46, 570 (1960); J. Tharrats, Compt. rend. 250, 3786 (1960); H. \Vakita, Progr. Theoret. Phys. 23, 32 (1960); 27, 139 (1962); \V. \Weidlich, Z. Naturforsch 15a, 651 lated by the attempts to alter the probabilistic interpretation of quantum mechanics. However, even when these attempts turned out to be less fruitful than its protagonists had hoped,’ the interest continued. Hence, after the subject had been dormant for more than two decades, we again hear discussions on the basic principles of quantum theory and the epistemologies that are compatible with it. As is often the case under similar circumstances, some of the early thinking had been forgotten; in fact, a small fraction ofGoogle Scholar
- 1.See also the articles of E. Teller, M. Born, A. Landé, F. Bopp, and G. Ludwig in Werner Heisenberg und die Physik unserer Zeit ( Friedrich Vieweg und Sohn, Braunschweig, 1961 ).Google Scholar
- 2.See the comments of V. Fock in the Max Planck Festschrift (Deutscher Verlag der Wissenschaften, Berlin, 1958), p. 177, particularly Sec. II.Google Scholar
- 3.W. Heisenberg, L. Physik 43, 172 (1927), also his article in Niels Bohr and the Development of Physics (Pergamon Press, London, 1955); N. Bohr, Nature 121, 580 (1928); Naturwissenschaften 17, 483 (1929) and par¬ticularly Atomic Physics and Human Knowledge (John Wiley and Sons, Inc., New York, 1958 ).Google Scholar
- 4.See J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Verlag Julius Springer, Berlin, 1932), English translation by the Princeton University Press, Princeton, New Jersey, 1955. See also P. Jordan, Anschau-.liche Quantentheorie (Julius Springer, Berlin, 1936), Chapter V.Google Scholar
- 5.F. London and E. Bauer, La Théorie de l’observation en mécanique quantique“ (Hermann et Cie., Paris, 1939); or E. Schrödinger, Naturwissenschaften 23, 807 ff. (1935); ”’roc. Cambridge Phil. Soc. 31, 555 (1935). whether we physicists do not go beyond our competence when searching for philosophical truth. I believe that we probably do.’ Neverthe¬less, the ultimate implications of quantum theory’s formulation of the laws of physics appear interesting even if one admits that the conclusions to be arrived at may not be the ultimate truth.Google Scholar
- 6.This point is particularly well expressed by H. Mar-genau, in the first two sections of the article in Phil. Sci. 25, 23 (1958).Google Scholar
- 7.The self-adjoint (Hermitean) character of every ob¬servable can be derived from Eq. (1) and the unitary nature of the transformation indicated by the arrow. Cf. E. Wigner, Z. Physik 133, 101 (1952), footnote 2 on p. 102.Google Scholar
- 8.N. F. Mott, Proc. Roy. Soc. (London) 126, 79 (1929).ADSCrossRefzbMATHGoogle Scholar
- 9.The same experiment was discussed recently from another point of view by H. \Cakita, Progr. Theoret. Phys. 27, 139 (1962).CrossRefGoogle Scholar
- 10.This point is disregarded by several authors who have rediscovered von Neumann’s description of the measure¬ment, as given by (1) and (2). These authors assume that it follows from the macroscopic nature of the measuring apparatus that if several values of the “pointer position” have finite probabilities [as is the case if the state vector is (2)], the state is necessarily a mixture (rather than a linear combination) of the states (3)—that is, of states in each of which the pointer position is definite (sharp). The argument given is that classical mechanics applies to macroscopic objects, and states such as (2) have nocounterpart in classical theory. This argument is contrary to present quantum-mechanical theory. It is true that the motion of a macroscopic body can be adequately described by the classical equations of motion if its state has a classical description. That this last premise is, according to present theory, not fulfilled, is clearly, though in an ex¬treme fashion, demonstrated by Schrödinger’s cat-paradox (cf. reference 5). Further, the discussion of the Stern-Gerlach experiment, given in the text, illustrates the fact that there are, in principle, observable differences between the state vector given by the right side of (2), and the mixture of the states (3), each of which has a definite position. Proposals to modify the quantum-mechanical equations of notion so as to permit a mixture of the states (3) to be the result of the measurement even though the initial state was a state vector, will be touched upon later.Google Scholar
- 11.See G. Ludwig’s article “Solved and Unsolved Prob¬lems in the Quantum Mechanics of Measurement” (refer¬ence I) and the present author’s article in The Scientist Speculates, edited by J. Good ( William Heinemann, Lon¬don, 1962 ), p. 284.Google Scholar
- 12.There are, nevertheless, other procedures to bring a system into a definite state. These are based on the fact that a small system, if it interacts with a large system in a definite and well-known state, may assume itself a definite state with almost absolute certainty. Thus, a hydrogen atom, in some state of excitation, if placed into a large container with no radiation in it, will almost surely transfer all its energy to the radiation field of the container and gc. over into its normal state. This method of preparing a state has been particularly stressed by H. MIargenau.Google Scholar
- 13.H. Araki and M. Yanase, Phys. Rev. 120, 666 (1961); cf. also E. P. \Vigner, Z. Physik 131, 101 (1952).Google Scholar
- 14.This point was recognized already by D. Bohm. See Section 22.11 of his Qaantunc Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1951 ).Google Scholar
- 15.C. E. Porter and N. Rosenzweig, Suomalaisen Tiedeakatemian Toimotuksia VI, No. 44 (1960); Phys. Rev. 120, 1698 (1960); F. Dyson, J. Math. Phys. 3, 140, 157, 166 (1962). Sec also E. P. Wigner, Proceedings of the Fourth Canadian Mathematics Congress ( University of Toronto Press, Toronto, 1959 ), p. 174.Google Scholar
- 16.Cf. the writer’s article in The Logic of Personal Knowl¬edge ( Routledge and Kegan Paul, London, 1961 ), p. 231.Google Scholar
- 17.N. Bohr and L. Rosenfeld, Kgl. Danske Videnskab. Selskab, Mat.-fvs. Medd. 12, No. 8 (1933); Phys. Rev. 78, 194 (1950); E. Corinaldesi, Nuovo cimento 8, 494 (1951); B. Ferretti, ibid. 12, 558 (1954).Google Scholar
- 18.See, in this connection, the rather similar situation discussed by A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).CrossRefGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1997