popper-logic-scientific-discovery

some observations on quantum theory 233
Heisenberg formulae can be statistically interpreted, and therefore (2)
that their interpretation as limitations upon attainable precision does
not follow logically from the quantum theory, which therefore could
not be contradicted merely by our attaining a higher degree of
precision in our measurements.* 1
‘So far, so good,’ someone might retort. ‘I won’t deny that it may be
possible to view quantum mechanics in this way. But it still does not
seem to me that the real physical core of Heisenberg’s theory, the
impossibility of making exact singular predictions, has even been touched
by your arguments’.
If asked to elaborate his thesis by means of a physical example, my
opponent might proceed as follows: ‘Imagine a beam of electrons, like
one in a cathode tube. Assume the direction of this beam to be the xdirection.
We can obtain various physical selections from this beam.
For example, we may select or separate a group of electrons according to
their position in the x-direction (i.e. according to their x-co-ordinates
at a certain instant); this could be done, perhaps, by means of a shutter
which we open for a very short time. In this way we should obtain a
group of electrons whose extension in the x-direction is very small.
According to the scatter relations, the momenta of the various electrons
of this group would differ widely in the x-direction (and therefore also
their energies). As you rightly stated, we can test such statements about
scattering. We can do this by measuring the momenta or the energies
of single electrons; and as we know the position, we shall thus obtain
both position and momentum. A measurement of this kind may be
carried out, for example, by letting the electrons impinge upon a plate
whose atoms they would excite: we shall then find, among other
things, some excited atoms whose excitation requires energy in excess
of the average energy of these electrons. Thus I admit that you were
quite right in stressing that such measurements are both possible and
significant. But—and now comes my objection—in making any such
measurement we must disturb the system we are examining, i.e. either
the single electrons, or if we measure many (as in our example), the
whole electron beam. Admittedly, the theory would not be logically
contradicted if we could know the momenta of the various electrons of
* 1 Point (3) of my programme has, in fact, been covered also.