Path-integral approach to the Schrodinger current - art. no. 052107

Discontinuous initial wave functions, or wave functions with a discontinuou s derivative and a bounded support, arise in a natural way in various situa tions in physics, in particular in measurement theory. The propagation of s uch initial wave functions is not well described by the Schrodinger current which vanishes on the boundary of the support of the wave function. This p ropagation gives rise to a unidirectional current at the boundary of the su pport. We use path integrals to define current and unidirectional current, and to provide a direct derivation of the expression for current from the p ath-integral formulation for both diffusion and quantum mechanics. Furtherm ore, we give an explicit asymptotic expression for the short-time propagati on of an initial wave function with compact support for cases of both a dis continuous derivative and a discontinuous wave function. We show that in th e former case the probability propagated across the boundary of the support in time Delta t is 0(Delta t(3/2)), and the initial unidirectional current is 0(Delta t(1/2)). This recovers the Zeno effect fur continuous detection of a particle in a given domain. For the latter case the probability propa gated across the boundary of the support in time Delta t is 0(Delta t(1/2)) , and the initial unidirectional current is 0(Delta t(-1/2)). This is an an ti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is 0(Delta t). This g ives a decay law.