FOUNDATIONS OF QUANTUM MECHANICS

80 CHAPTER IV. THE COPENHAGEN INTERPRETATION
The more closely the position is determined, δq is small, the more inaccurately the momentum afterwards
is known, δp is large.
Quoting Heisenberg again (loc. cit.)
At the instant when position is determined - therefore, at the moment when the photon is
scattered by the electron - the electron undergoes a discontinuous change in momentum.
This change is the greater the smaller the wavelength of the light employed - that is, the
more exact the determination of the position. At the instant at which the position of the
electron is known, its momentum therefore can be known up to magnitudes which correspond
to that discontinuous change. Thus, the more precisely the position is determined,
the less precisely the momentum is known, and conversely.
This conclusion is the first formulation of the uncertainty principle. According to Heisenberg’s
own measuring = defining principle this conclusion can, however, not yet be drawn because it also
has to be specified what, in this context, must be understood by the term ‘momentum of the electron’.
In a later discussion (Heisenberg 1930), Heisenberg specifies the reasoning by also discussing the
definition of the momentum of the electron.
This reasoning goes as follows. Suppose that the momentum of the electron has been measured
in advance with an inaccuracy δ p 1 . Next, the position is measured with an inaccuracy δ q, then the
momentum is measured again, with inaccuracy δp 2 . We can assume that δp 1 ≪ p 1 and δp 2 ≪ p 2 ,
so that the momentum is very accurately known before and after the position measurement. Now it
makes sense to speak of the momentum p 1 of the electron shortly before the position measurement.
If now the position is measured very precisely, the position and momentum of the electron in the past
are arbitrarily well defined. Heisenberg (1930, p. 20):
[. . . ] if the velocity of the electron is at first known and the position then exactly measured,
the position for times previous to the measurement may be calculated. Then for
these past times δp δq is smaller than the usual limiting value [. . . ]
Apparently, the uncertainty relation does not apply to the past. In the example the uncertainty concerns
the unpredictability of the value of p 2 after the position measurement, not the inaccuracy δp 2
with which p 2 can be measured. This unpredictability can be determined by accurately measuring
the momentum before and after the determination of position, and the unpredictability is larger if
the determination of position was more precise. Although it is true that one can speak in a logically
consistent manner of the position and momentum of the electron in the past (loc. cit.),
[. . . ] but this knowledge of the past is of a purely speculative character, since it can never
(because of the unknown change in momentum caused by the position measurement) be
used as an initial condition in any calculation of the future progress of the electron and
thus cannot be subjected to experimental verification. It is a matter of personal belief
whether such a calculation concerning the past history of the electron can be ascribed
any physical reality or not.
For Heisenberg, such a calculation does not describe reality. But then, what is reality to him?
Heisenberg says, (1927, Eng. tr. p. 73),
The “orbit” comes into being only when we observe it.