This study investigates three-dimensional quantum tunneling through analytical solutions of the Schr¨odinger equation in Cartesian and spherical coordinates. By develop ing transmission coefficients for both square and spherical potential barriers, we establish a theoretical framework for α-decay via the Gamow model. The analysis reveals how angular momentum and energy dependence govern tunneling probabilities, with experimental valida tion showing agreement within 15% for high-energy decays. The results bridge fundamental quantum mechanics with nuclear physics applications.
This study investigates three-dimensional quantum tunneling through analytical solutions of the Schr¨odinger equation in Cartesian and spherical coordinates. By develop ing transmission coefficients for both square and spherical potential barriers, we establish a theoretical framework for α-decay via the Gamow model. The analysis reveals how angular momentum and energy dependence govern tunneling probabilities, with experimental valida tion showing agreement within 15% for high-energy decays. The results bridge fundamental quantum mechanics with nuclear physics applications.
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.
Quantum tunnelling represents one of the most fundamental conse-quences of quantum mechanics. The phenomenon allows particles to pen-etrate potential barriers even when their kinetic energy is insufficient ac-cording to classical mechanics. Although initially discovered while study-ing nuclear decay, tunnelling has since emerged as a unifying mechanism across multiple areas of physics. In astrophysics, tunnelling enables nu-clear fusion inside stellar cores by allowing positively charged nuclei to penetrate the electrostatic Coulomb barrier. This process provides the primary energy source for stars and enables the cosmic synthesis of heavy elements necessary for planetary systems and biological life. The proba-bility of such reactions is governed by the Gamow factor and the resulting Gamow peak in stellar energy distributions. Beyond astrophysics, tun-nelling phenomena play a central role in modern quantum technologies, including scanning tunnelling microscopy, semiconductor devices, and su-perconducting Josephson junctions used in quantum computing architec-tures. In theoretical physics, tunnelling also appears in models of Hawk-ing radiation, providing a quantum interpretation for particle emission from black holes. This work presents a unified theoretical exploration of quantum tunnelling across multiple physical scales. Beginning with the Schrödinger equation and barrier penetration theory, we examine tun-nelling in nuclear astrophysics, stellar reaction rates, quantum devices, and gravitational systems. Finally, we discuss emerging informational perspectives in modern physics, suggesting that probabilistic quantum processes may reveal deeper informational structures underlying physical reality.
According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is known to be. This event is called tunneling. Tunneling is a probability, and it is measured by a transition coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In this paper, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values.
Quantum tunneling — the passage of a particle through a potential barrier that it classically lacks the energy to surmount — has been described since Gamow (1928) by the tunneling probability T ≈ exp(−2κd), where κ = √(2m(V₀−E))/ℏ and d is the barrier width. The formula is well-confirmed experimentally but has not been derived from a physical principle more fundamental than the Schrödinger equation. We derive it from the principle of information processing load minimization, using the framework established in the preceding papers of this series. We define the tunneling load as C_tunnel = ∫κ(x)dx — the integral of the spatial attenuation coefficient through the barrier — and show that T = exp(−2·C_tunnel) exactly. This is not a restatement of the WKB approximation; it is an identification of the physical meaning of the integral ∫κ(x)dx as the information processing load of the tunneling path. The tunneling path is selected by load minimization: when C_tunnel < C_classical (the load of the classical over-barrier trajectory), the system tunnels. Three physical systems are verified numerically without free parameters: (i) proton-proton nuclear fusion in the solar core — Gamow load C_Gamow = πη = 11.28, T = 1.60 × 10⁻¹⁰, consistent with solar luminosity; (ii) scanning tunneling microscope (STM) current — C_STM = κd = 5.43 at 0.5 nm, T = 1.91 × 10⁻⁵, and the factor-of-ten current change per 0.1 nm distance change follows directly; (iii) Ra-226 alpha decay — C_Gamow = 79.2, T ~ 10⁻⁶⁹, consistent with the 1,600-year half-life. Tunneling is not a mysterious violation of classical mechanics. It is the selection of the minimum-load propagation path — the same principle that explains wave-particle duality (Paper 7), wave function collapse (Paper 4), and the structure of all fundamental forces (Paper 8).
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
The present work sought to deduce the alpha decay theory based on quantum tunneling and using approximations in the resolution of the Schrödinger equation for Coulomb potential. It was possible to find the alpha decay constant analytically. One of the approaches used was the WKB method for solving differential equations, by which the Gamow factor was reached. From this point on, it was possible to identify the dependence of the decay rate with the energy of the radiation, as well as the atomic number. Besides that, we could understand the behavior of the tunneling probability with respect to some variables such as emitted particle energy and atomic number of the nucleus that decays. With this research we can help researchers in general in the study of alpha radiation.