A Path Intergal Approach to Current

Abstract

Discontinuous initial wave functions or wave functions with discontintuous derivative and with bounded support arise in a natural way in various situations in physics, in particular in measurement theory. The propagation of such initial wave functions is not well described by the Schrödinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a uni-directional current at the boundary of the support. We use path integrals to define current and uni-directional current and give a direct derivation of the expression for current from the path integral formulation for both diffusion and quantum mechanics. Furthermore, we give an explicit asymptotic expression for the short time propagation of initial wave function with compact support for both the cases of discontinuous derivative and discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $\Delta t$ is $O(\Delta t^{3/2})$ and the initial uni-directional current is $O(\Delta t^{1/2})$. This recovers the Zeno effect for continuous detection of a particle in a given domain. For the latter case the probability propagated across the boundary of the support in time $\Delta t$ is $O(\Delta t^{1/2})$ and the initial uni-directional current is $O(\Delta t^{-1/2})$. This is an anti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is $O(\Delta t)$. This gives a decay law.

17 pages, Latex

References in corpus (3)

• Measurement of Time-of-Arrival in Quantum Mechanics
• Quantum anti-Zeno effect
• Feynman Integrals with Absorbing Boundaries

Cited by in corpus (15)

• Quantum Zeno effect by general measurements
• Quantum Zeno and anti-Zeno effects in the Friedrichs model
• Brownian Simulations and Uni-Directional Flux in Diffusion
• Pitfalls of Path Integrals: Amplitudes for Spacetime Regions and the Quantum Zeno Effect
• Analytical study of quadratic and non-quadratic short-time behavior of quantum decay
• Can the double-slit experiment distinguish between quantum interpretations?
• Universal potential barrier penetration by initially confined wavepackets
• Measurement as Absorption of Feynman Trajectories: Collapse of the Wave Function Can be Avoided
• Classical Limit of the Quantum Zeno Effect by Environmental Decoherence
• Schroedinger current for discontinuous states from the first passage time decomposition
• Time dispersion in quantum electrodynamics
• Does the Heisenberg uncertainty principle apply along the time dimension?
• Bound Eigenstate dynamics under a sudden shift of the well's wall
• Time dispersion in bound states
• Inferring the smoothness of the autocorrelation function from that of the initial state