popper-logic-scientific-discovery

306
appendices
slit. This angle φ can be made as small as we like by making |a 2|
sufficiently large; for we have
(1)
h
φ ≅
r.|a2| where r is the width of the slit. But it is impossible by this method to
decrease|∆a 2|; it would decrease only by increasing r which would
lead to an increase of|∆P|; for we have
(2)
which in view of (1) leads to
(3)
|∆a 2|≅ φ|a 2|
|∆a 2|≅ h
r
showing that|∆a 2|does not depend upon|a 2|.
Owing to the fact that for any chosen r we can make φ as small as we
like by increasing |a 2|, we can also make the component ∆a 2 in the
PX-direction, which we denote by ‘(∆a 2) x’, as small as we like; and we
can do this without interfering with the precision of the measurement
of the position of P, since this position too becomes more precise with
increasing |a 2|, and decreasing r. Now we wish to show that a
corresponding argument holds for (∆b 2) y, i.e. for the PY-component of
∆b 2.
Since we may put ∆a 1 = 0 (according to our assumption), we obtain
from the conservation of momenta
(4)
∆b 2 = ∆b 1 − ∆a 2
For any given a 1, |b 1| and |a 2|, ∆b 1 depends directly upon φ,
which means that we can have an arrangement such that
(5)
|∆b 1| ≅ |∆a 2| ≅ h
r