BOHMIAN ANALYSIS OF MOMENTUM-TRANSFER IN WELCHER WEG MEASUREMENTS

Whether the loss of visibility due to measurement in two-slit interfer ence patterns can always be attributed to momentum transfer depends up on how one defines the last term. The momentum transfer to a quantum p article can be determined unambiguously if it was initially in a momen tum eigenstate, but that is not the state relevant to a two-slit exper iment. A sensible answer for the two-slit problem was obtained by Wise man ct nl. [Phys. Rev. A 56, 944 (1997)] using the formalism of the Wi gner 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-total(B)(p') for it to receive a momentum t ransfer p' as a result of the measurement. Furthermore, the 1-norm of this distribution obeys the relation [\p'\](total)(B)d greater than or equal to 2 (h) over bar(1 - V)/pi, where V is the visibility of the i nterference pattern and d is the slit separation. This confirms that t he momentum transfer in a welcher Weg (which path) experiment is in ac cord with the uncertainty principle. Like the Wignerian analysis, the Bohmian analysis clearly distinguishes between a local momentum transf er (which occurs even if the particle is localized at a single slit) a nd 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 mea surement, and is a consequence of Bohm's ''quantum potential.''