Weak and semiweak values in non-ideal spin measurements: an exact treatment beyond the asymptotic regime

We consider weak measurements (WM) of a spin observable in quantum mechanics beyond the usual asymptotic regime. This is done by obtaining the exact time-dependent wave functions of the measuring apparatus for general non-ideal measurements. Ideal strong measurements and the usual WM regime are obtained as the two extreme limiting cases of our exact treatment. We show that in the intermediate regime non-ideal measurements lead to "semiweak" eccentric values that differ from the usual weak values. We further show that even in the WM regime the exact treatment leads to a meter behavior that can be markedly distinct from the one predicted by the usual WM formalism. We give several illustrations and discuss an application to the distinction of different realizations of the same density matrix.