Bohr explicitly rejected the idea that measurements could be seen as disturbing certain definite properties of the system. Instead of viewing the uncertainty principle as reflecting an uncontrollable aspect of the measurement process, he saw it as expressing a limit to thedefinability of concepts such as position and momentum:
Uncertainty and complementarity
2.3
the uncertainty in question is not simply a consequence of a discontinu- ous change of energy and momentum say during an interaction between radiation and material particles employed in measuring the space-time coordinates of the individuals [. . . ] the question is rather that of the impossibility of defining rigorously such a change when the space-time coordination of the individuals is also considered. (Bohr, 1985, p. 93)
The background of this view rests on yet another principle, namely, that of com- plementarity. According to this principle, ‘position of a particle’ and ‘momentum of a particle’ refer to mutually incompatible modes of description of a system. Fur- thermore, neither mode of description provides an exhaustive description: both are needed to obtain a full characterization of the system.
The fact that Bohr was not entirely satisfied with Heisenberg’s exposition of the uncertainty principle was brought to Heisenberg’s attention just before his paper was published. The clash resulted in a small “addition in proof” in the paper. In this addition it is not entirely clear what Heisenberg’s position on Bohr’s complaint was, but in a later work he reflects on the period between finishing his paper and publishing it as one in which they reached agreement:
After several weeks of discussion, which were not devoid of stress, we soon concluded, not least thanks to Oskar Klein’s participation, that we really meant the same, and that the uncertainty relations were just a special case of the more general complementarity principle. Thus, I sent my improved paper to the printer and Bohr prepared a detailed publication on complementarity. (Heisenberg, 1967, p. 105)
It is, however, not entirely clear how the uncertainty principle can be viewed as a special case of the complementarity principle. After all, the uncertainty principle links two mutually exclusive modes of description into a single relation. The com- plementarity principle, if taken seriously, should preclude a meaningful reading of this relation. Such a view can indeed be found in the work of Bohr (1949, p. 211): “It must here be remembered that even in the indeterminacy relation we are dealing with an implication of the formalism which defies unambiguous expression in words suited to describe classical pictures.” On this reading, the uncertainty principle only makes sense in the limit cases where either position (∆x = 0) or momentum (∆p= 0) is well-defined.
This view is often rejected though, and the uncertainty relation is asserted to be a meaningful mathematical expression. The reason for this may possibly be traced back to the idea, upheld by many physicists, that the wave function or quantum state expresses something real. For example, Born writes
We regard waves on a lake as real, though they are nothing material but only a certain shape of the surface of the water. The justification is that
they can be characterized by certain invariant quantities, like frequency and wavelength, or a spectrum of these. Now the same holds for light waves; why then should we withhold the epithet ‘real’, even if the waves represent in quantum theory only a distribution of probability? (Born, 1953, p. 149)
To see why this view on quantum waves is relevant, it needs to be noted that the uncertainty principle not only occurs as an empirical principle, but also comes in a version that is a theorem of quantum mechanics itself. In fact, this version of the principle (due to Kennard (1927)) is more widely known, and discussed in any textbook. Instead of considering possible experimental setups of position and momentum measurements, this principle is directly derived from the theory. To be precise, it states that for every quantum state ψ the variances in position X and momentumP satisfy the inequality
Varψ(X)Varψ(P)≥~/2. (2.1)
Here X and P are now operators on a Hilbert space, ~denotes Planck’s constant,
and the variance is given by the Born rule.
Besides the shift from a pretheoretical principle to a theorem of a theory, there is another important distinction between this relation and the one obtained by Heisenberg. In Heisenberg’s relation the uncertainties refer to the precision of actual measurements. The inequality (2.1), on the other hand, uses Born’s interpretation of the wave function and identifies the uncertainties with spreads in the probability distributions associated withψ. If this wave function is real in some sense, then it seems at least plausible that the associated probability distributions are also real. The uncertainty principle then is not only meaningful in the limit cases, but also expresses a definite feature of the world in any other case. This is, of course, in strong contrast with the view on the quantum formalism held by Bohr:7
The entire formalism is to be considered as a tool for deriving pre- dictions, of definite or statistical character, as regards information ob- tainable under experimental conditions described in classical terms [. . . ] These symbols themselves [. . . ] are not susceptible to pictorial interpre- tation; and even derived real functions like densities and currents are only to be regarded as expressing the probabilities for the occurrence of individual events observable under well-defined experimental conditions. (Bohr, 1948, p. 314)
7It deserves to be noted though that Born himself at many points also sided with Bohr’s phi-
losophy. Even though for him the quantum state was perfectly real, the associated probability distributions only have meaning with respect to an experimental setup: “The prediction made by the theory [. . . ] has a meaning only in relation to the whole experimental arrangement” (Born, 1953, p. 146).
Reflection
2.4
There are several things that may be taken away from these considerations. First, Born’s rule for calculating probabilities from the quantum state was widely accepted. But second, there wasn’t much consensus on the ontological status of these states or their associated probabilities. With Born one sees the strongest ten- dency to view quantum probabilities as inherent aspects of reality. In the writing of Bohr one finds a more modest interpretation, associating quantum probabilities with descriptions of experiments rather than that they define the mechanisms be- hind these experiments. And finally, in the writing of Einstein, one finds the view that quantum mechanics is at best a preliminary theory. The ontological status of quantum probabilities can then only be understood after a more complete theory has been devised.