popper-logic-scientific-discovery

appendix vi 303
This assumption, however, is incompatible with the quantum theory as at present
accepted. For although the theory permits discontinuous jumps, it permits
them only in the case of particles within an atom (within the
range of discontinuous Eigen-values, but not in the case of free particles
within the range of continuous Eigen-values).
It is possible, presumably, to design a theory (in order to escape the
conclusions reached above, or to preserve the principle of indeterminacy)
which alters the quantum theory in such a way that the assumption
of a disturbance of the position by selecting the momentum is
compatible with it; but even this theory—which I might call a ‘theory
of indeterminacy’—could derive only statistical consequences from
the principle of indeterminacy, and could therefore be corroborated
only statistically. Within this theory, the principle of indeterminacy
would only be a formally singular probability statement, although its
content would go beyond what I have called the ‘statistical scatter
relations’. For, as will be shown below with the help of an example,
these are compatible with the assumption that selecting the
momentum does not disturb the position. Thus this latter assumption does not
allow us to infer the existence of a ‘super-pure case’ such as is forbidden by the scatter
relations. This statement shows that the method of measuring I have
examined does not affect the statistically interpreted formulae of
Heisenberg. It may thus be said to occupy, within my statistical interpretation,
the same ‘logical place’, as it were, as (within his interpretation)
Heisenberg’s statement denying the ‘physical reality’ of exact
measurements; in fact one might regard my statement as the translation
of Heisenberg’s statement into the statistical language.
That the statement in question is correct may be seen from the
following considerations. We might try to obtain a ‘super-pure case’ by
reversing the order of the steps in the experiment; by first selecting,
say, a position in the x-direction (the flight direction) with the help of
a fast shutter, and only afterwards selecting the momentum with the
help of a filter. This might be thought feasible; for as a result of the
position-measurement, all sorts of momenta would appear, out of
which the filter—without disturbing the position—will select only
those which happened to fall within some small range. But these considerations
would be mistaken. For if a group of particles is selected by
an ‘instantaneous shutter’, in the way indicated, then Schrödinger’s