Uncertainty principle for joint measurement of noncommuting variables

Does the Heisenberg uncertainty principle refer to errors of measurement, or to the spread of values of the physical variables intrinsic to a particle’s state, or to some combination of these? The most commonly quoted form of the uncertainty principle relates the spread of an ensemble of separate measurements of some variable to the analogous spread of its conjugate variable. In contrast, Heisenberg’s original argument for the uncertainty principle involved the perturbation to a particle’s state by a measurement of one variable, which affects one’s ability to predict the outcome of a subsequent measurement of the conjugate variable. The relation between these two views of the uncertainty principle is discussed in this paper. A familiar example is considered: an ensemble of identically prepared particles passing through a slit, and after further propagation being detected. From this arrangement it is possible to infer joint (although necessarily imprecise) information on both transverse position and conjugate momentum for each member of the ensemble. It is shown that in this case of joint measurement the product of standard deviations for the measurement outcomes is at least twice as large as the lower bound implied by the usual uncertainty principle. The discussion is meant to help clarify the different roles played in the various statements of the uncertainty principle by the initial state and by measurement error.