Variance of the quantum dwell time for a nonrelativistic particle

Abstract

Muñoz, Seidel, and Muga [Phys. Rev. A 79, 012108 (2009)10.1103/PhysRevA.79.012108], following an earlier proposal by Pollak and Miller [Phys. Rev. Lett. 53, 115 (1984)10.1103/PhysRevLett.53.115] in the context of a theory of a collinear chemical reaction, showed that suitable moments of a two-flux correlation function could be manipulated to yield expressions for the mean quantum dwell time and mean square quantum dwell time for a structureless particle scattering from a time-independent potential energy field between two parallel lines in a two-dimensional spacetime. The present work proposes a generalization to a charged, nonrelativistic particle scattering from a transient, spatially confined electromagnetic vector potential in four-dimensional spacetime. The geometry of the spacetime domain is that of the slab between a pair of parallel planes, in particular, those defined by constant values of the third (z) spatial coordinate. The mean Nth power, N = 1, 2, 3, …, of the quantum dwell time in the slab is given by an expression involving an N-flux-correlation function. All these means are shown to be nonnegative. The N = 1 formula reduces to an S-matrix result published previously [G. E. Hahne, J. Phys. A 36, 7149 (2003)10.1088/0305-4470/36/25/316]; an explicit formula for N = 2, and of the variance of the dwell time in terms of the S-matrix, is worked out. A formula representing an incommensurability principle between variances of the output-minus-input flux of a pair of dynamical variables (such as the particle's time flux and others) is derived.