popper-logic-scientific-discovery

some observations on quantum theory 237
know of any of the particles in which direction it will turn after passing
through the slit; but if we consider one definite direction we can calculate
precisely the momentum component of all particles that did turn
in this particular direction. Thus the particles which after having passed
through the slit travel in one definite direction again form an imagined
selection. We are able to predict their position and their momentum, or
in short, their paths; and again, by putting a photographic plate in their
path, we can test our predictions.
The situation is in principle the same (even though empirical tests
are somewhat more difficult) in the case of the first example we considered,
namely the selection of particles according to their position in
the direction of travel. If we produce a physical selection corresponding
to this case, then different particles will travel with different velocities,
because of the spread of the momenta. The group of particles will
thus spread over an increasing range in the x-direction as it proceeds.
(The packet will get wider.) We can then work out the momentum of a
partial group of these particles (selected in imagination) which, in a
given moment, will be at a given position in the x-direction: the
momentum will be the greater the farther ahead is the selected partial
group (and vice versa). The empirical test of the prediction made in this
way could be carried out by substituting for the photographic plate a
moving strip of photographic film. As we could know of each point in
the band the time of its exposure to the impact of the electrons we
could also predict for each point on the band with what momentum the
impacts would occur. These predictions we could test, for example by
inserting a filter in front of the moving band or perhaps in front of the
Geiger-counter (a filter in the case of light rays; in the case of electrons
an electric field at right angles to the direction of the ray) followed by a
selection according to direction, allowing only those particles to pass
which possess a given minimum momentum. We could then ascertain
whether these particles really did arrive at the predicted time or not.
The precision of the measurements involved in these tests is not
limited by the uncertainty relations. These are meant to apply, as we
have seen, mainly to those measurements which are used for the
deduction of predictions, and not for testing them. They are meant to
apply, that is to say, to ‘predictive measurements’ rather than to ‘non-predictive
measurements’. In sections 73 and 76 I examined three cases of such