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

Thu Feb 22 09:58:31 MET 1996