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Let us consider the simplest case: that of a two-state classical system. We call its two states "off" and "on." Its action is simple: if it is off, then it will stay off forever. If it is on, then it can detect a particle and go off. Later on we will specialize to detection of particle presence at a given location in space. For a while let us be general and assume that we have two Hilbert spaces tex2html_wrap_inline560 and two Hamiltonians tex2html_wrap_inline562 . We also have a time dependent family of operators tex2html_wrap_inline564 and let us denote tex2html_wrap_inline566 . According to the theory presented in the previous section, with tex2html_wrap_inline568 , tex2html_wrap_inline570 , the master equation for the total system, i.e. for particle and detector, reads:


Suppose at t=0 the detector is "on" and the particle state is tex2html_wrap_inline574 , with tex2html_wrap_inline576 Then, according to the event generating algorithm described in the previous section, probability of detection during time interval (0,t) is equal to tex2html_wrap_inline580

Let us now specialize and consider a detector of particle present at a location a in space (of n dimensions). Our detector has a certain range of detection and certain efficiency. We encode these detector characteristics in a gaussian function:


If the detector is moving in space along some trajectory a(t), and if the detector characteristics are constant in time, then we put: tex2html_wrap_inline588 . Let us suppose that the detector is off at tex2html_wrap_inline590 and that the particle wave function is tex2html_wrap_inline592 . Then, according to the algorithm described in the previous section, probability of detection in the infinitesimal time interval tex2html_wrap_inline594 equals tex2html_wrap_inline596 . In the limit tex2html_wrap_inline598 , when tex2html_wrap_inline600 we get tex2html_wrap_inline602 . Thus we recover the usual Born interpretation, with the evident and necessary correction that the probability of detection is proportional to the length of exposure time of the detector.
That simple formula holds only for short exposure times. For a prolonged detection, the formula becomes more involved, primarily because of non-unitary evolution due to the presence of the detector. In that case, numerical simulation is necessary. To get an idea of what happens, let us consider a simplified case which can be solved exactly. We will consider the ultra-relativistic Hamiltonian H=-i d/dx in space of one dimension. In that case the non-unitary evolution equation is easily solved:


In the limit tex2html_wrap_inline598 when the detector shrinks to a point, and assuming that this point is fixed in space a(t)=a, we obtain for the probability p(t) of detecting the particle in the time interval (0,t):


Intuitively this result is very clear. Our Hamiltonian describes a particle moving to the right with velocity c=1, the shape of the wave packet is preserved. Then p(t) is equal to the probability that the particle at t=0 was in a region of space that guaranteed passing the detector, multiplied by the detector efficiency factor - in our case this factor is tex2html_wrap_inline620

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