In this chapter we show how the weak values, are related to the T0µ(x, t) component of the energy–momentum tensor. This enables the local energy and momentum to be measured using weak measurement techniques. We also show how the Bohm energy and momentum are related to T0µ(x, t) and therefore it follows that these quantities can also be measured using the same methods. Thus the Bohm “trajectories” can be empirically determined, as was shown by Kocsis et al. (2011a) in the case of photons. Because of the difficulties with the notion of a photon trajectory, we argue the case for determining experimentally similar trajectories for atoms where a trajectory does not cause these particular difficulties.

Introduction

The notion of weak measurement introduced by Aharonov, Albert and Vaidman (1988) and Aharonov and Vaidman (1990) has opened up a radically new way of exploring quantum phenomena. In contrast to the strong measurement (von Neumann, 1955), which involves the collapse of the wave function, a weak measurement induces a more subtle phase change which does not involve any collapse. This phase change can then be amplified and revealed in a subsequent strong measurement of a complementary operator that does not commute with the operator being measured. This amplification explains why it is possible for the result of a weak spin measurement of a spin-1/2 atom to be magnified by a factor of 100 (Aharonov et al., 1988; Duck, Stevenson and Sudarshan, 1989). A weak measurement, then, provides a means of amplifying small signals as well as allowing us to gain new, more subtle information about quantum systems.

One of the new features that we will concentrate on in this chapter is the possible measurement of the T0µ (x, t) components of the energy–momentum tensor.