popper-logic-scientific-discovery

302
appendices
interposition of a filter. (If the beam consists of electrons we shall have
to use instead of a filter an electric field perpendicular to the direction
of the beam in order to analyse its spectrum.) We assume with Heisenberg
that this procedure leaves unaltered the momenta (or, more precisely,
their components in the x-direction) and consequently also the
velocities (or their x-components) of the selected particles.
Behind the filter we put a Geiger-counter (or a moving strip of
photographic film) in order to measure the time of arrival of the
particles; and this allows us—since the velocities of the particles are
known—to calculate their x-co-ordinates for any instant preceding
their time of arrival. Now we may consider two possible assumptions.
If, on the one hand, it is assumed that the x-co-ordinates of the positions
of the particles were not interfered with by the measuring of their
momenta, then the measurement of position and momentum can be
validly extended to the time before the momentum was selected (by
the filter). If, on the other hand, it is assumed that a selection according
to the momentum does interfere with the x-co-ordinates of the positions
of the particles, then we can calculate their paths exactly only for
the time-interval between the two measurements.
Now the assumption that the position of the particles in the direction
of their flight might be disturbed in some unpredictable way by a
selection according to a given momentum means the same as that the
position co-ordinate of a particle would be altered in some incalculable
way by this selection. But since the velocity of the particle has remained
unchanged, this assumption must be equivalent to the assumption that,
owing to that selection, the particle must have jumped discontinuously
(with super-luminal velocity) to a different point of its path.
his argument. (In other words, I still believe that my argument and my experiment of
section 77 can be used to point out an inconsistency in Heisenberg’s discussion of the
observation of an electron.) But I now believe that I was wrong in assuming that what
holds for Heisenberg’s imaginary ‘observations’ or ‘measurements’ would also hold for
my ‘selections’. As Einstein shows (in appendix *xii), it does not hold for a filter acting
upon a photon. Nor does it hold for the electric field perpendicular to the direction of a
beam of electrons, mentioned (like the filter) in the first paragraph of the present appendix.
For the width of the beam must be considerable if the electrons are to move parallel
to the x-axis, and as a consequence, their position before their entry into the field cannot
be calculated with precision after they have been deflected by the field. This invalidates
the argument of this appendix and the next, and of section 77.