popper-logic-scientific-discovery

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some structural components of a theory of experience
measurements will generally yield a completely random distribution of
the positions in the x direction.)
Thus in their physical application, our statistical scatter relations
come to this. If one tries, by whatever physical means, to obtain as
homogeneous an aggregate of particles as possible, then this attempt will
encounter a definite barrier in these scatter-relations. For example, we
can obtain by means of physical selection a plane monochromatic
ray—say, a ray of electrons of equal momentum. But if we attempt to
make this aggregate of electrons still more homogeneous—perhaps by
screening off part of it—so as to obtain electrons which not only have
the same momentum but have also passed through some narrow slit
determining a positional range ∆x, then we are bound to fail. We fail
because any selection according to the position of the particles
amounts to an interference with the system which will result in
increased scattering of the momentum components p x, so that the scattering
will increase (in accordance with the law expressed by the
Heisenberg formula) with the narrowing of the slit. And conversely: if
we are given a ray selected according to position by being passed
through a slit, and if we try to make it ‘parallel’ (or ‘plane’) and
monochromatic, then we have to destroy the selection according to
position since we cannot avoid increasing the width of the ray. (In the
ideal case—for example, if the p x, components of the particles are all to
become equal to o—the width would have to become infinite.) If the
homogeneity of a selection has been increased as far as possible (i.e. as
far as the Heisenberg formulae permit, so that the sign of equality in
these formulae becomes valid) then this selection may be called a pure
case. 5
Using this terminology, we can formulate the statistical scatter
5 The term is due to Weyl (Zeitschrift fur Physik 46, 1927, p. 1) and J. von Neumann
(Göttinger Nachrichten, 1927, p. 245). If, following Weyl (Gruppentheorie und Quantenmechanik,
p. 70; English translation p. 79; cf. also Born-Jordan, Elementare Quanten-mechanik,
p. 315), we characterize the pure case as one ‘. . . which it is impossible to produce
by a combination of two statistical collections different from it’, then pure cases
satisfying this description need not be pure momentum or place selections. They could
be produced, for example, if a place-selection were effected with some chosen degree
of precision, and the momentum with the greatest precision still attainable.