popper-logic-scientific-discovery

some observations on quantum theory 223
relations thus: There is no aggregate of particles more homogeneous
than a pure case.* 2
It has not till now been taken sufficiently into account that to the
mathematical derivation of the Heisenberg formulae from the fundamental
equations of quantum theory there must correspond, precisely,
a derivation of the interpretation of the Heisenberg formulae from the
interpretation of these fundamental equations. March for instance has
described the situation just the other way round (as indicated in the
previous section): the statistical interpretation of quantum theory
appears in his presentation as a consequence of the Heisenberg limitation
upon attainable precision. Weyl on the other hand gives a strict
derivation of the Heisenberg formulae from the wave equation—an
equation which he interprets in statistical terms. Yet he interprets the
Heisenberg formulae—which he has just derived from a statistically
interpreted premise—as limitations upon attainable precision. And he
does so in spite of the fact that he notices that this interpretation of the
formulae runs counter in some respects to the statistical interpretation
of Born. For according to Weyl, Born’s interpretation is subject to ‘a
correction’ in the light of the uncertainty relations. ‘It is not merely the
case that position and velocity of a particle are just subject to statistical
laws, while being precisely determined in every single case. Rather, the
very meaning of these concepts depends on the measurements needed
to ascertain them; and an exact measurement of the position robs us of
the possibility of ascertaining the velocity.’ 6
The conflict perceived by Weyl between Born’s statistical interpretation
of quantum theory and Heisenberg’s limitations upon attainable
precision does indeed exist; but it is sharper than Weyl thinks. Not only
is it impossible to derive the limitations of attainable precision from
the statistically interpreted wave-equation, but the fact (which I have
still to demonstrate) that neither the possible experiments nor the
actual experimental results agree with Heisenberg’s interpretation can
* 2 In the sense of note *1, this should, of course, be re-formulated: ‘There is no experimental
arrangement capable of producing an aggregate or sequence of experiments with
results more homogeneous than a pure case.’
6
Weyl, Gruppentheorie und Quantenmechanik, p. 68. *The paragraph here cited seems to be
omitted in the English translation.