FINDING: Quantum tunneling is a wave-mechanical penetration of finite potential barriers, with the transmission probability governed by exponential decay of the wavefunction inside the barrier; recent QFT work shows "impossible" measurements are mathematically possible but non-ideal. | MATH: Transmission coefficient T≈e−2κL for rectangular barrier, where κ=2m(V0−E)/ℏ; decay length δ=1/κ=ℏ/2m(V0−E). For arbitrary barrier V(x), WKB approximation: T≈exp(−2∫x1x22m(V(x)−E)/ℏdx). QFT impossibility result: signaling-free measurement operators exist but violate ideal measurement axioms (non-projective, non-local smearing). | CONNECTION: The exponential decay factor e−2κL is structurally identical to the golden-ratio-related decay in Fibonacci-like recursive barrier stacks — for N identical barriers, transmission resonances occur when kL=nπ, a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
A simple solvable model of two-channel inelastic scattering is constructed. The contact-type interaction is realized mathematically by a self-adjoint extension; if the interchannel interaction is switched off it reduces to one point interaction in each channel. The scattering matrix is calculated and the bound states and resonances are found in the weak-coupling case. The decay of the resonance state is also considered; it is shown that the asymptotic behaviour of the non-exponential corrections to the decay law depends substantially on the spectrum of the unperturbed system
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.