Niels Bohr: Complementary principle (1885–1962)

     Quantum 
                 Mechanics 

Niels Bohr introduced and explained his concept of “complementarity” in his famous 1927 Como Lecture (reproduced in Bohr 1934). He recognized the need for the mathematical formalism of quantum mechanics to be imbedded a rationally coherent conceptual framework if it were to serve as the core of an acceptable scientific theory. Yet the applications of the formalism were based upon the integration of two logically incompatible conceptual structures, the mathematical formalisms of classical and quantum physics. The applications that we normally make of quantum theory involve three physical systems: (1), the system being examined; (2), the measuring devices by means of which we probe its properties; and (3), our own physical bodies. All three systems are composed of atoms, and hence must be describable in terms of the mathematical concepts of quantum theory. Yet our observations are described in terms of the contents of our sense experiences, which, for the phenomena under consideration, are described in terms of the concepts of classical physics.

Classical physics postulates that, at each instant of time, each elementary particle is located at some definite point in space, and has a definite velocity, and hence also a definite momentum. On the other hand, in quantum mechanics an elementary particle is represented by a distribution of possibilities, where the distributions in position and in momentum are related by Fourier transformation. This entails that localization at a point in position space demands a complete lack of localization in momentum space, and vice versa. Bohr associates “causation” with the law of conservation of momentum and energy, and hence is able to say that:

The very nature of quantum theory thus forces us to regard the claim of space-time co-ordination and the claim of causality, the union of which characterizes the classical theories, as complementary but exclusive features of the description, symbolizing the idealization of observation and definition respectively. (Bohr 1934, p. 54)

Bohr explains that:

The quantum theory is characterized by the acknowledgement of a fundamental limitation in the classical physical ideas when applied to atomic phenomena.    …its essence may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity, or rather individuality, completely foreign to classical theories and symbolized by Planck’s quantum of action.…the quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected.  Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation. After all, the concept of observation is in so far arbitrary as it depends upon which objects are included in the system to be observed. Ultimately, every observation can, of course, be reduced to our sense perceptions.”  (Bohr 1934, p. 53)

These passages gives a glimpse of the range and complexity of the ideas that Bohr wants to integrate into his rationally coherent foundation for the application and use of quantum theory.

The elaboration that he provides in the remainder of the Como lecture is lengthy, but its essence is summarized and updated in his 1958 paper “Quantum physics and Philosophy: Causality and Complementarity”, in which he says:

Within the scope of classical physics, all characteristic properties of a given object can in principle be ascertained by a single experimental arrangement, although in practice various arrangements are often convenient for the study of different aspects of the phenomena. In fact, data obtained in such a way simply supplement each other and can be combined into a consistent picture of the behaviour of the object under investigation. In quantum mechanics, however, evidence about atomic objects obtained by different experimental arrangements exhibits a novel kind of complementary relationship. Indeed, it must be recognized that such evidence which appears contradictory when combination into a single picture is attempted, exhaust all conceivable knowledge about the object. Far from restricting our efforts to put questions to nature in the form of experiments, the notion of complementarity simply characterizes the answers we can receive by such inquiry, whenever the interaction between the measuring instruments and the objects form an integral part of the phenomena. 

(Bohr 1962, p.4)

Compactly stated, the essential idea here is that in quantum theory the information provided by different experimental procedures that in principle cannot, because of the physical character of the needed apparatus, be performed simultaneously, cannot be represented by any mathematically allowed quantum state of the system being examined. The elements of information obtainable from incompatible measurements are said to be complementary: taken together exhaust the information obtainable about the state. On the other hand, any preparation protocol that is maximally complete, in the sense that all the procedures are mutually compatible and are such that no further procedure can add any more information, can be represented by a quantum state, and that state represents in a mathematical form all the conceivable knowledge about the object that experiments can reveal to us.

As regards the closely connected issue of causality, Bohr says:

In the treatment of atomic problems, actual calculations are most conveniently carried out with the help of a Schroedinger state function, from which the statistical laws governing observations obtainable under specified conditions can be deduced by definite mathematical operations. It must be recognized, however, that we are dealing here with a purely symbolic procedure, the unambiguous physical interpretation of which in the last resort requires reference to the complete experimental arrangement. (Bohr 1963, p. 5)

This relegation of the Schroedinger state function, which gives the space-time representation of the atomic substrate of all systems, to a purely symbolic status, might seem to be denigrating this Schroedinger representation of the state relative to others. But the point is rather that it puts the Schroedinger space-time representation on a par with the others:

In fact, wave mechanics, just as the matrix theory, represents on this view a symbolic transcription of the problem of motion of classical mechanics adapted to the requirements of quantum theory and only to be interpreted by an explicit use of the quantum postulate. (Bohr 1934, p.75)

All of this must be understood within the basic pragmatic premise of Bohr’s approach:

In our description of nature the purpose is not to disclose the real essence of phenomena but only to track down as far as possible relations between the multifold aspects of our experience. (Bohr 1934, p. 18)

References: 

Bohr, N. (1934): Atomic theory and the description of nature. Cambridge University Press, Cambridge UK.

Bohr, N. (1963): Essays 1958-1962 on Atomic Physics and Human Knowledge. Wiley, New York.