FOUNDATIONS OF QUANTUM MECHANICS

IV. 2. BOHR AND COMPLEMENTARITY 85
The opposite applies to the measurement of momentum (Bohr in Schilpp 1949, p. 219);
In the study of phenomena in the account of which we are dealing with detailed momentum
balance, certain parts of the whole device must naturally be given the freedom to
move independently of others.
Bohr assumes that a measurement of momentum is made by registering the recoil after a collision,
for example, with a test particle. In this way we can, using the conservation laws, retrieve the
momentum of the object. However, the condition that the test particle can move freely means that we
cannot guarantee that it preserves a definite position. It is therefore excluded from being used as part
of a spatial coordinate system, and now we cannot say anything about the position of the object.
In order to perform a position measurement we must therefore put the object in contact with a
part of the measuring apparatus which has been bolted down firmly, while performing a momentum
measurement we must observe the recoil of a freely movable part of the measuring apparatus, and
apply the momentum conservation law. Position and momentum measurements therefore exclude
each other, because a measuring apparatus cannot at the same time be bolted down and freely movable.
In the description of the object we must choose between granting a position or momentum. As worded
by Philipp Frank (1949, p. 163)
Quantum mechanics speaks neither of particles the positions and velocities of which
exist but cannot be accurately observed, nor of particles with indefinite positions and
velocities. Rather, it speaks of experimental arrangements in the description of which the
expressions ”position of a particle” and ”velocity of a particle” can never be employed
simultaneously.
Bohr calls this characteristic property of quantum mechanics, where two quantities exclude each
other whereas both are necessary to describe all phenomena in which the object can participate, complementarity.
Position and momentum are examples of complementary quantities. Similar considerations
apply to time and energy, such that a general complementarity exists between on the one hand
a space - time description of phenomena, and on the other hand a dynamical description, frequently
indicated by Bohr as ‘causally’, in which the conservation laws for energy momentum are applicable.
◃ Remark
The complementarity between quantities like position and momentum or descriptions using space -
time coordination or dynamic laws differs from, and replaces, the contrast which Bohr placed central
in his earlier work, namely between ‘wave’ and ‘particle’, because a classical particle has both position
and momentum, a classical wave has neither. ▹
The role of the uncertainty relations in Bohr’s views can now be described as considering them
in the first place as symbolic expressions of the impossibility to define position and momentum at
the same time when describing an object. In a phenomenon in which the position is determined
sharply, δ q = 0, the momentum must be undetermined, δ p = ∞, and vice versa. But the relation
δq δp ∼ h is, of course, more general. Bohr (1934, pp. 60,61) interprets this as follows:
At the same time, however, the general character of this relation makes it possible to
a certain extent to reconcile the conservation laws with the space - time co - ordination
of observations, the idea of a coincidence of well - defined events in a space - time point
being replaced by that of unsharply defined individuals within finite space - time regions.