Analysis of quantum decay law: Is quantum tunneling really exponential?

TL;DR

This paper investigates whether quantum decay truly follows an exponential law by solving the Schrödinger equation for specific potentials and analyzing the survival probability, revealing non-exponential behavior at certain times.

Contribution

It provides an analytical and numerical study demonstrating deviations from exponential decay in quantum systems, challenging the traditional decay law.

Findings

Non-exponential decay observed at short and intermediate times.

Analytical solutions for specific potentials obtained.

Numerical simulations confirm non-exponential behavior.

Abstract

The exponential decay law is well established since its first derivation in 1928, however it is not exact but only an approximate description. In recent years some experimental and theoretical indications for non-exponential decay have been documented. First we solve analytically the time-dependent Schr\"odinger equation in one dimension for a potential consisting of an infinite wall plus a rectangular barrier with finite width and also a cut harmonic oscillator potential by considering it as a sequence of square potentials. Then using the staggered Leap-Frog method, we solve the time-dependent Schr\"odinger equation for the cut harmonic oscillator potential. In both methods, time dependence of the survival probability of the particle and the decay parameter {\lambda} are analyzed. The results exhibit non-exponential behavior for survival probability at short and intermediate times.