[Paper Review] Wave functions and characteristic times for transmission and reflection

N. L. Chuprikov|ArXiv.org|May 6, 2004

Quantum optics and atomic interactions3 references3 citations

TL;DR

This paper proposes a novel wave-packet analysis framework in which quantum tunneling through a one-dimensional potential barrier is treated as a combined process of two elementary, causally distinct sub-processes: transmission and reflection. By deriving exact and asymptotic wave functions for each, the authors introduce unique, physically grounded characteristic times for transmission and reflection, resolving ambiguities in the standard wave-packet approach and providing a causal interpretation of tunneling dynamics.

ABSTRACT

We present a renewed wave-packet analysis based on the following ideas: if a quantum one-particle scattering process and the corresponding state are described by an indivisible wave packet to move as a whole at all stages of scattering, then they are elementary; otherwise, they are combined; each combined process consists from several alternative elementary ones to proceed simultaneously; the corresponding (normed) state can be uniquely presented as the sum of elementary ones whose (constant) norms give unit, in sum; Born's formula intended for calculating the {\it expectation} values of physical observables, as well as the standard timing procedure are valid only for elementary states and processes; only an elementary time-dependent state can be considered as the quantum counterpart to some classical one-particle rajectory. By our approach, tunneling a non-relativistic particle through a static one-dimensional potential barrier is a combined process consisting from two elementary ones, transmission and reflection. In the standard setting of the problem, we find an unique pair of solutions to the Schrödinger equation, which describe separately transmission and reflection. On this basis we introduce (exact and asymptotic) characteristic times for transmission and reflection.

Motivation & Objective

• To resolve the long-standing tunneling time problem (TTP) by redefining the temporal description of quantum scattering.
• To address the inconsistency in standard wave-packet analysis (SWPA), where the center-of-mass motion does not causally link incident, transmitted, and reflected packets.
• To establish that tunneling is a combined process composed of two elementary, non-interfering sub-processes: transmission and reflection.
• To provide a physically consistent basis for defining characteristic times by focusing on individual elementary states rather than the total wave packet.

Proposed method

• Introduces a new formalism where the full scattering process is decomposed into two elementary, norm-conserving components: transmission and reflection.
• Derives exact and asymptotic solutions to the time-dependent Schrödinger equation that describe transmission and reflection as independent processes.
• Applies Born’s formula and standard timing procedures only to these elementary states, ensuring causal and unambiguous time definitions.
• Defines characteristic times (e.g., dwell, phase, or arrival times) based on the individual wave functions for transmission and reflection, not the total wave packet.
• Uses symmetry and spatial localization of external probes (e.g., magnetic fields or absorbing potentials) to test the separation of processes experimentally.
• Extends the framework to general one-dimensional potentials, including asymmetric barriers, potential steps, and spherically symmetric scattering.

Experimental results

Research questions

• RQ1 Why does the standard wave-packet approach fail to provide a consistent definition of tunneling time, and what is the root cause of this failure?
• RQ2 Can the non-causal behavior of transmitted wave packets (e.g., superluminal group velocities) be explained by the interference of multiple processes rather than intrinsic dynamics?
• RQ3 What is the correct physical basis for defining characteristic times in tunneling, and how can they be uniquely assigned to transmission and reflection?
• RQ4 How can the wave function be decomposed into elementary components that each represent a causally closed scattering process?
• RQ5 To what extent can experimental probes like Larmor precession or time-of-arrival measurements be reinterpreted in terms of separate transmission and reflection wave functions?

Key findings

• The tunneling process is fundamentally a combined process composed of two alternative elementary processes: transmission and reflection, each with its own independent wave function.
• The standard wave-packet approach fails because it treats the total wave packet as a single causal entity, while the transmitted and reflected components are not causally connected to the incident packet.
• Exact and asymptotic wave functions for transmission and reflection are derived, which are norm-conserving and allow for unambiguous timing procedures.
• Characteristic times for transmission and reflection are uniquely defined and differ from standard phase or dwell times, providing a more physically grounded basis.
• The Hartman effect and superluminal group velocities are explained not as particle acceleration, but as artifacts of interference in a combined process.
• The framework is generalizable to arbitrary one-dimensional potentials, including asymmetric barriers, potential steps, and spherically symmetric scattering.