The uncertainty relation in 'which-way' experiments: how to observe directly the momentum transfer using weak values

A which-way measurement destroys the twin-slit interference pattern. Bohr argued that this can be attributed to the Heisenberg uncertainty relation: distinguishing between two slits a distance s apart gives the particle a random momentum transfer (sic) 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), on the other hand, proved a theorem that, they claimed, showed that Bohr was right. This work reviews and extends a recent proposal (Wiseman 2003 Phys. Lett. A 311285) to resolve the issue using a weak-valued probability distribution for momentum transfer, P-wv(sic). We show that P-wv(sic) must be nonzero for some (sic) : \(sic)\ > h/6s. However, its moments can be identically zero, such as in the experiment proposed by SEW. This is possible because P-wv (sic) is not necessarily positive definite. Nevertheless, it is measurable experimentally in a way understandable to a classical physicist. The new results in this paper include the following. We introduce a new measure of spread for P-wv (sic): half the length of the unit-confidence interval. We conjecture that it is never less than h/4s, and find numerically that it is approximately h/1.59s for an idealized version of the SEW scheme with infinitely narrow slits. For this example, the moments of P-wv (sic), and of the momentum distributions, are undefined unless a process of apodization is used. However, we show that by considering successively smoother initial wavefunctions, successively more moments of both P-wv (sic) and the momentum distributions become defined. For this example the moments of P-wv (sic) are zero, and these moments are equal to the changes in the moments of the momentum distribution. We prove that this relation also holds for schemes in which the moments of P-wv (sic) are nonzero, but it holds only for the first two moments. We also compare these moments to the moments of two other momentum-transfer distributions that have previously been considered, and with the moments of (p) over capf-(p) over capi (which is defined in the Heisenberg picture). We find agreement between all of these, but again only for the first two moments. Our results reconcile the seemingly opposing views of SEW and STCW.