popper-logic-scientific-discovery

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some structural components of a theory of experience
uncertainty relations. It is argued that, owing to this uncertainty of measurements
in any atomic experiments, ‘. . . the result will not in general
be determinate, i.e. if the experiment is repeated several times under
identical conditions several different results may be obtained. If the
experiment is repeated a large number of times it will be found that
each particular result will be obtained in a definite fraction of the total
number of times, so that one can say there is a definite probability of its
being obtained any time the experiment is performed’ (Dirac). 3
March too writes with reference to the uncertainty relation: ‘Between
the present and the future there hold . . . only probability relations;
from which it becomes clear that the character of the new mechanics
must be that of a statistical theory.’ 4
I do not think that this analysis of the relations between the
uncertainty formulae and the statistical interpretation of the quantum
theory is acceptable. It seems to me that the logical relation is just the
other way round. For we can derive the uncertainty formulae from
Schrödinger’s wave equation (which is to be interpreted statistically),
but not this latter from the uncertainty formulae. If we are to take due
account of these relations of derivability, then the interpretation of the
uncertainty formulae will have to be revised.
75 A STATISTICAL RE-INTERPRETATION OF
THE UNCERTAINTY FORMULAE
Since Heisenberg it is accepted as an established fact that any simultaneous
measurements of position and momentum with a precision
exceeding that permitted by his uncertainty relations would contradict
quantum theory. The ‘prohibition’ of exact measurements, it is
believed, can be logically derived from quantum theory, or from wave
3 Dirac, Quantum Mechanics, 1930, p. 10. *(From the 1st edition.) A parallel passage,
slightly more emphatic, occurs on p. 14 of the 3rd edition. ‘. . . in general the result will
not be determinate, i.e., if the experiment is repeated several times under identical
conditions several different results may be obtained. It is a law of nature, though, that if
the experiment is repeated a large number of times, each particular result will be
obtained in a definite fraction of the total number of times, so that there is a definite
probability of its being obtained.’
4 March, Die Grundlagen der Quantenmechanik, p. 3.