popper-logic-scientific-discovery

holds, and therefore also
(6)
(7)
|∆b 1| − |∆a 2| ≅ h
r
Moreover, we obtain, in analogy to (2),
|∆b 2| ≅ ψ.|b 2|,
where ‘ψ’ denotes the indeterminacy of the direction of b 2. Accordingly
we obtain in view of (4) and (5)
(8)
ψ ≅ |∆b 1 − ∆a 2|
b 2
h
≅
r.|b2| ;
appendix vii 307
But this means: however small we make r, we always can make ψ and
with it (∆b 2) y as small as we like by using sufficiently high values for
the momentum |b 2|; and this, again, without interfering with the
precision of the measurement of the position P.
This shows that it is possible to make each of the two factors of the
product (∆P) y.(∆b 2) y as small as we like, independently of the other.
But for the refutation of Heisenberg’s assertion as to the limits of
attainable precision, it would have been sufficient to show that one of
these two factors can be made as small as we like without making the
other grow beyond all bounds.
In addition it may be noted that by an appropriate choice of the PXdirection
it is possible to determine the distance PX in such a way that ∆P
and ∆b 2 are parallel, and thus (for sufficiently small φ) normal to PY. 1
In consequence, the precision of the momentum in this direction, and
moreover the precision of the position (in the same direction), both
become independent of the precision of the measurement of the position of P. (The
1 The fact that an examination of the degree of the exactness of measurement taken in a
direction perpendicular to ∆s can be relevant, was pointed out to me by Schiff during a
discussion of my imaginary experiment.
I wish to offer here my warmest thanks to Dr. K. Schiff for fruitfully collaborating
throughout the better part of a year.