The notion of the weak value of a quantum mechanical observable has recently been introduced by Aharonov et al ., hereafter referred to as AAV. Their analysis led to the controversial result that the weak values for an observable can lie well outside its associated eigenvalue spectrum. We have performed an experiment similar to one proposed by Knight et al . and demonstrated that weak values can be observed and that they exhibit the properties described by AAV within the regime of validity defined by Duck et al . It is shown that these properties are produced by interference phenomena that result from the loss of 'welcher Weg' information. We have also identified an intrinsic 'weak energy' which appears in the equation of motion for the weak value of an observable and have shown that it can be non-vanishing only for systems which are both preselected and postselected. It is noted that such a 'weak spin-rotation coupling energy' can be associated with our experiment and that it is manifested in our experimental data. We suggest that preselection and postselection measurement techniques could be useful for the creation and study of physical phenomena that might be otherwise unobservable.

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The quantum theoretical concepts of modular momentum and dynamical non-locality, which were introduced four decades ago, have recently been used to explain single particle quantum interference phenomena. Although the non-local exchange of modular momentum associated with such phenomena cannot be directly observed, it has been suggested that effects induced by this exchange can be measured experimentally using weak measurements of pre-and post-selected ensembles of particles. This paper reports on such an optical experiment that yielded measured weak values that were consistent with the theoretical prediction of an effect induced by a non-local exchange of modular momentum.

The variance of an arbitrary pointer observable is considered for the general case that a complex weak value is measured using a complex valued pointer state. For the typical cases where the pointer observable is either its position or momentum, the associated expressions for the pointer's variance after the measurement contain a term proportional to the product of the weak value's imaginary part with the rate of change of the third central moment of position relative to the initial pointer state just prior to the time of the measurement interaction when position is the observable-or with the initial pointer state's third central moment of momentum when momentum is the observable. These terms provide a means for controlling pointer position and momentum variance and identify control conditions which, when satisfied, can yield variances that are smaller after the measurement than they were before the measurement. Measurement sensitivities which are useful for estimating weak-value measurement accuracies are also briefly discussed.