popper-logic-scientific-discovery

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some structural components of a theory of experience
travelling in the x-direction. But this time we assume it to be monochromatic,
so that all the particles are travelling along parallel paths in
the x-direction with the same known momentum. The components in
the other directions of the momentum will then also be known, that is,
known to be equal to zero. Now instead of determining the position in
the x-direction of a group of particles by means of a physical selection—
instead, that is, of isolating the group of particles from the rest of the
beam by technical means (as we did above)—we shall be content to
differentiate this group from the rest merely by focusing our attention
upon it. For example, we may focus our attention upon all those particles
which have (with a given precision) in a given instant the place
co-ordinate x, and which therefore do not spread beyond an arbitrarily
small range ∆x. Of each of these particles we know the momentum
precisely. We therefore know for each future instant precisely where
this group of particles is going to be. (It is clear that the mere existence
of such a group of particles does not contradict quantum theory; only
its separate existence, that is, the possibility of selecting it physically,
would contradict the theory.) We can carry out the same kind of
imaginary selection in connection with the other space co-ordinates.
The physically selected monochromatic beam would have to be very
wide in the y and z-directions (infinitely wide in the case of an ideal
monochromatic beam) because in these directions the momentum is
supposed to be selected with precision, i.e. to be equal to o; so that
positions in these directions must be widely spread. Nevertheless we
can again focus our attention upon a very narrow partial ray. Again, we
shall not only know the position but also the momentum of every
particle of this ray. We shall therefore be able to predict for every
particle of this narrow ray (which we have, as it were, selected in
imagination) at which point, and with what momentum, it will
impinge upon a photographic plate set in its path, and of course we can
test this prediction empirically (as with the former experiment).
Imaginary selections, analogous to the one just made from a ‘pure
case’ of a particular type, can be made from other types of aggregates.
For example, we may take a monochromatic beam from which a physical
selection has been made by means of a very small slit ∆y (thus
taking as our physical starting point a physical selection corresponding
to the merely imagined selection of the preceding example). We do not