popper-logic-scientific-discovery

affected, for example by irradiation with light. It is thus impossible to
infer from the result of the measurement the precise state of an atomic
object immediately after it has been measured. Therefore the measurement
cannot serve as basis for predictions. Admittedly, it is always possible to ascertain,
by means of new measurements, the state of the object after the
previous measurement, but the system is thereby again interfered with
in an incalculable way. And admittedly, it is always possible to arrange
our measurements in such a way that certain of the characteristics of
the state to be measured—for example the momentum of the
particle—are not disturbed. But this can only be done at the price of
interfering the more severely with certain other characteristic magnitudes
of the state to be measured (in this case the position of the
particle). If two magnitudes are mutually correlated in this way then
the theorem holds for them that they cannot simultaneously be measured
with precision, although each may be separately so measured.
Thus if we increase the precision of one of the two measurements—say
the momentum p x, thereby reducing the range or interval of error
∆p x—then we are bound to decrease the precision of the measurement
of the position co-ordinate x, i.e. to expand the interval ∆x. In this way,
the greatest precision attainable is, according to Heisenberg, limited by
the uncertainty relation, 2
∆x . ∆p x � h
4π .
Similar relations hold for the other co-ordinates. The formula tells us
that the product of the two ranges of error is at least of the order of
magnitude of h, where h is Planck’s quantum of action. It follows from
this formula that a completely precise measurement of one of the two
magnitudes will have to be purchased at the price of complete
indeterminacy in the other.
According to Heisenberg’s uncertainty relations, every measurement
of the position interferes with the measurement of the corresponding
component of the momentum. Thus it is in principle impossible to
2 For the derivation of this formula cf. note 2 to section 75.
some observations on quantum theory 213