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Example: Dirac's counter for ultra-relativistic particle

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.gif 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

Arkadiusz Jadczyk
Thu Feb 22 09:58:31 MET 1996