Let us now specialize the model by assuming that we consider a particle
in 
and that the Hilbert space vector |a> approaches the
improper position eigenvector 
localized at the point a.
This corresponds to a point--like detector of strength 
placed at a.
We see from the
equation (4) that 
is in this case given by:
where the complex amplitude 
of the particle arriving at a is:
or, from Eq. (12)
where
stands for the Laplace transform of 
.
Let us now consider the simplest explicitly solvable example - that of an
ultra--relativistic particle on a line. For 
we take 
then the propagator 
is given by 
and its
Laplace transform reads 
. In
particular 
and from Eq. (26) we
see that the amplitude for arriving at the point a is given by the "almost
evident" formula:
where
It follows that probability that the particle will be registered is equal to
which has a maximum 
for 
if the support of 
is left to the counter position 
We notice that in this
example the
shape of the arrival time probability distribution 
does not depend
on the value of the coupling constant - only the effectiveness of the
detector depends on it. For a counter corresponding to a superposition
we obtain for 
exactly the
same expression as for one counter but with 
replaced with
