Numerical Studies of Non-Exponential Decay of Wavefunctions

A. Petridis, Jon Vermedahl, L. P. Staunton +1 more

A. Petridis, L. P. Staunton, Jon Vermedahl +1 more

The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact solutions are obtained by means of a superposition of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions for which the resulting integrals can be evaluated exactly. Square-integrability and the boundary conditions are satisfied. The simplest of the obtained solutions is presented and the probability for the particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential decay for all times; the probability decreases at large times as . Other exact solutions found exhibit power law behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally, numerical solutions are obtained using the staggered leap-frog algorithm for select potentials exhibiting the prevalence of non-exponential decay at short times.

和夫 斉藤

A. J. O'Connor

Andrew Stewart Caldin

FINDING: Quantum tunneling is a wave-mechanical penetration of classically forbidden barriers, governed by exponential decay of the wavefunction; "impossible" QFT measurements are possible but non-ideal, with signaling constraints. | MATH: Transmission coefficient \(T \approx e^{-2\kappa L}\), where \(\kappa = \sqrt{2m(V_0-E)}/\hbar\); barrier penetration probability \(P \propto e^{-2\gamma}\), \(\gamma = \frac{1}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\). For rectangular barrier: \(T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0-E)}}\). QFT impossibility: measurement operators \(\hat{M}\) with \(\langle \psi | [\hat{M}(x), \hat{O}(y)] | \psi \rangle \neq 0\) for spacelike \(x-y\) lead to superluminal signaling unless \(\hat{M}\) is "ideal" (projection-valued, local). | CONNECTION: The exponential decay constant \(\kappa L\) is dimensionless — when \(e^{-2\kappa L}\) equals the golden ratio inverse \(0.618\), we get \(\kappa L = \frac{1}{2}\ln(1/\varphi) \approx 0.2406\); wh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

K.J.F. Gaemers, T.D. Visser