  A which-way measurement destroys the twin-slit interference pattern. Bohr
argued that distinguishing between two slits a distance s apart gives the
particle a random momentum transfer \wp of order h/s. This was accepted for
more than 60 years, until Scully, Englert and Walther (SEW) proposed a
which-way scheme that, they claimed, entailed no momentum transfer. Storey,
Tan, Collett and Walls (STCW) in turn proved a theorem that, they claimed,
showed that Bohr was right. This work reviews and extends a recent proposal
[Wiseman, Phys. Lett. A 311, 285 (2003)] to resolve the issue using a
weak-valued probability distribution for momentum transfer, P_wv(\wp). We show
that P_wv(\wp) must be wider than h/6s. However, its moments can still be zero
because P_wv(\wp) is not necessarily positive definite. Nevertheless, it is
measurable in a way understandable to a classical physicist. We introduce a new
measure of spread for P_wv(\wp): half of the unit-confidence interval, and
conjecture that it is never less than h/4s. For an idealized example with
infinitely narrow slits, the moments of P_wv(\wp) and of the momentum
distributions are undefined unless a process of apodization is used. We show
that by considering successively smoother initial wave functions, successively
more moments of both P_wv(\wp) and the momentum distributions become defined.
For this example the moments of P_wv(\wp) are zero, and these are equal to the
changes in the moments of the momentum distribution. We prove that this
relation holds for schemes in which the moments of P_wv(\wp) are non-zero, but
only for the first two moments. We also compare these moments to those of two
other momentum-transfer distributions and \hat{p}_f-\hat{p}_i. We find
agreement between all of these, but again only for the first two moments.
