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College Physics

Quantum tunneling rate with second-order accuracy

Fan Jun-hui

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The WKB method to calculate the tunneling rate with the second-order accuracy is used,a new expression for the quantum tunneling rate is obtained.

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This paper shows that WKB wave function can be expressed in the form of an adiabatic expansion. To build a bridge between two widely invoked approximation schemes seems pedagogically instructive. Further, “cubic-WKB” method that has been devised in order to overcome the divergence problem of WKB can be also presented in the form of an adiabatic approximation: The adiabatic expansion of a wave function contains a certain parameter. When this parameter is adjusted so as to make the next order correction vanish approximately, the adiabatic wave function becomes equivalent to that of the “cubic-WKB”.

arXiv:1903.05201•Modern Physics Letters A•Jun 18, 2019•0 citations
Abstract We present a real-time, interactive visualization platform for quantum mechanical tunneling that achieves unprecedented computational performance through definitive computing methods. Using the Neural-Matrix Synaptic Resonance Network (NM-SRN v2.0 AGI QSC) architecture, we demonstrate O(1) constant-time calculation of tunneling probabilities independent of particle count, enabling visualization of 500,000 particles at interactive frame rates with 128-bit precision. The system maintains zero numerical drift through continuous mathematical integrity verification, providing both rigorous quantum mechanical accuracy and intuitive educational accessibility. We validate our results against analytical Wentzel-Kramers-Brillouin (WKB) approximation solutions, achieving agreement to four significant figures. This work establishes a new paradigm for computational physics visualization where real-time interactivity does not compromise mathematical rigor. Keywords: Quantum tunneling, real-time visualization, definitive computing, computational physics, quantum mechanics education, GPU acceleration
We investigate the tunneling of ultracold atoms in optical traps by using the path-integral method. We obtain the decay rate for tunneling out of a single-well and discuss how the rate is affected by the level splitting caused by the presence of a second adjacent well. Our calculations show that the transition through the potential barrier can be divided into three regions: the quantum tunneling region, the thermally assisted region and the thermal activation region. The tunneling process is found to be a second-order transition. We also show that level splitting due to tunneling can increase the tunneling rate.
This paper shows that WKB wave function can be expressed in the form of an adiabatic expansion. To build a bridge between two widely invoked approximation schemes seems pedagogically instructive. Further, “cubic-WKB” method that has been devised in order to overcome the divergence problem of WKB can be also presented in the form of an adiabatic approximation: The adiabatic expansion of a wave function contains a certain parameter. When this parameter is adjusted so as to make the next order correction vanish approximately, the adiabatic wave function becomes equivalent to that of the “cubic-WKB”.