# Tunneling time and superposition principle

## TL;DR

This paper demonstrates that tunneling and scattering processes violate the superposition principle, leading to a nonlinear wave model that resolves the tunneling time paradox and challenges traditional linear frameworks in quantum mechanics and electrodynamics.

## Contribution

It introduces a nonlinear wave model for tunneling and scattering, revising the superposition principle to address longstanding issues like the Hartman paradox.

## Key findings

- Violation of superposition principle in tunneling and scattering
- Development of a nonlinear wave model for quantum tunneling
- Resolution of the Hartman paradox in tunneling time

## Abstract

We show that scattering a quantum particle on a one-dimensional potential barrier as well as scattering the electromagnetic wave on a quasi-one-dimensional layered structure (both represent scattering problems with one 'source' and two 'sinks') violate the superposition principle; the role of nonlinear elements is played here by the potential barrier and the layered structure, splitting the incident (probability and electromagnetic) wave into two parts (transmitted and reflected). This explains why all attempts to solve the tunneling time problem within the framework of the standard (linear) models of these processes, both in quantum mechanics and in classical electrodynamics, have been unsuccessful. We revise the traditional formulation of the superposition principle, present a new (nonlinear) wave model, by the example of the quantum-mechanical scattering process, and show that concepts of the tunneling time developed on its basis are free from the Hartman paradox.