Whether the loss of visibility due to measurement in two-slit interference patterns can always be attributed to momentum transfer depends upon how one defines the last term. The momentum transfer to a quantum particle can be determined unambiguously if it was initially in a momentum eigenstate, but that is not the state relevant to a two-slit experiment. A sensible answer for the two-slit problem was obtained by Wiseman et al. [Phys. Rev. A 56, 944 (1997)] using the formalism of the Wigner function. Here I show that a more general answer can be obtained using the Bohmian formulation of quantum mechanics, in which particles have a definite position and momentum at all times. By following all possible trajectories of the particle it is possible to calculate the probability distribution ${P}_{\mathrm{total}}^{B}{(p}^{\ensuremath{'}})$ for it to receive a momentum transfer ${p}^{\ensuremath{'}}$ as a result of the measurement. Furthermore, the 1-norm of this distribution obeys the relation $〈|{p}^{\ensuremath{'}}|{〉}_{\mathrm{total}}^{B}d>~2\ensuremath{\Elzxh}(1\ensuremath{-}V)/\ensuremath{\pi},$ where
V is the visibility of the interference pattern and
d is the slit separation. This confirms that the momentum transfer in a welcher Weg (which path) experiment is in accord with the uncertainty principle. Like the Wignerian analysis, the Bohmian analysis clearly distinguishes between a local momentum transfer (which occurs even if the particle is localized at a single slit) and a nonlocal momentum transfer (which occurs only if the particle is delocalized, at both slits). In the Bohmian analysis the former occurs at the time of the measurement, while the latter occurs after the measurement, and is a consequence of Bohm's ``quantum potential.''
Abstract We construct proper microscopic propositions to consistently explain the interference patterns obtained for electrons and photons in well-defined one- and two-slit experiments. Our propositions are also made explicit on the basis of quantum mechanics with proper quantitative analysis to support the correctness of the physical mechanisms proposed herein, which obey the wave-particle duality. Apart from that, the microscopic propositions constructed herein can be treated as guidelines in setting up the future experiments such that one can unambiguously support or oppose this strange notion—a quantum particle behaves differently depending upon the number of slit, and where we place the detector such that the particle can sense the detector and the number of slit, before physically interacting with them.