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.
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.
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.
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.
La radiactividad es el proceso de desintegración y transformación espontánea de núcleos atómicos volátiles, seguido de la emisión de radiación. George Gamow fue capaz de utilizar la mecánica cuántica para explicar el fenómeno radiactivo del decaimiento alfa a través de un proceso llamado tunelización. Las partículas α son estructuralmente equivalentes al núcleo de un átomo de Helio y constan de dos protones y dos neutrones. Básicamente el argumento es que una partícula alfa no necesita tener suficiente energía para superar la barrera potencial del núcleo, simplemente tunelea a través de él. En este artículo se hace una revisión a los conceptos y se desentrañan los cálculos que utilizó Gamow para poder explicar su teoría de decaimiento. Palabras Clave: Decaimiento alfa, radiactividad, tunelamiento cuántico, mecánica cuántica. Abstract Radioactivity is the process of spontaneous decay and transformation of volatile atomic nuclei, followed by the emission of radiation. George Gamow was able to use quantum mechanics to explain the radioactive phenomenon of alpha decay through a process called tunneling. The α-particles are structurally equivalent to the nucleus of a helium atom and consist of two protons and two neutrons. Basically the argument is that an alpha particle does not need enough energy to overcome the potential barrier of the nucleus, it tunnels through it. This article reviews the concepts and unravels the calculations that Gamow used to explain his theory of decay. Keywords: Alpha decay, radioactivity, quantum tunneling, quantum mechanics. Referencias [1] Leon van Dommelen. Quantum Mechanics for Engineers, Version 5.55 alpha. 2012. [2] G. Gamow. Quantum Theory of the Atomic Nucleus. 1928. [3] David J. Griffiths. Introduction to Quantum Mechanics (2nd Edition). 2nd. Pearson Prentice Hall. isbn: 0131118927. [4] Michael F L’Annunziata. Radioactivity: introduction and history, from the quantum to quarks; 2nd ed. Amsterdam: Elsevier, 2016. url: https : / / cds . cern . ch / record / 2210635. [5] W.E. Lee M.I. Ojovan. An Introduction to Nuclear Waste Immobilisation; 2nd ed. Elsevier, 2014. [6] A C Melissinos and Jim Napolitano. Experiments in modern physics; 2nd ed. New York, NY: Academic Press, 2003. url: https://cds.cern.ch/ record/897801. [7] Eugen Merzbacher. “The Early History of Quantum Tunneling”. In: Physics Today 55.8 (Aug. 2002), pp. 44–49. doi: 10.1063/1.1510281. url: https://doi.org/10. 1063/1.1510281. [8] Curt A. Moyer Raymond A. Serway Clement J. Moses. Modern Physics, Third Edition. David Harris, 2005. [9] Thaddeus J. Trenn. “Rutherford on the AlphaBetaGamma Classification of Radioactive Rays”. In: Isis 67.1 (1976), pp. 61–75. issn: 00211753, 15456994. url: http: //www.jstor.org/ stable/231134.
In (1), an outline of the derivation of the Gamow factor for alpha decay from a nucleus is given. In one dimension, the approach considers three regions in space, one with an incident and reflected free particle wavefunction, the second (within a square V barrier) with an exponentially growing and falling solution, and the third, with a single free particle wavefunction exp(-iEt)exp(ipx+phase) representing the escaped alpha particle. Alpha decay is treated as a quantum tunneling problem and the wavefunction and its first derivative d/dx must be continuous. A second step in (1), however, is to assume that one may write a time related wavefunction as exp(- g t), where d/dt partial Integral dx (region) W*W = ½ (-i W* d/dx W + i W d/dx W*). This may appear to be a quantum mechanical equation, but we argue that this is nothing more than the classical continuity equation. In fact, in (2) we argued that quantum mechanics does not exist without classical equations. In particular, the quantum free particle wavefunctions exp(-iEt) and exp(ipx) follow exp(iEt)exp(-iEt) = 1 and exp(-ipx)exp(ipx) = 1 to link them classical densities in time and space for particles moving with a constant speed. If there is a decay in time and space, then the appropriate classical equation is the continuity one. Writing d/dt partial d(t) = constant d(t) yields the familiar exp(-g t) form. We apply the ideas of the Gamow factor to a set of photons bouncing back and forth in an n1 1-dimensional cavity surrounded by n2 (n2>n1) index of refraction material. In this case, if the photons are synchronized then one may follow them classically. The catch is that the probability to reflect or refract is quantum. In general, one would not write exp(-g t) in such a scenario, but would formally write an equation of the form: fraction of remaining photons = R power Integer(t/t1), where R is the probability to reflect and integer(t/t1) is the number of cycles.. If, however, the probability to refract is very small, we show that one may in fact obtain the same result as in the case of exp(- g t) for g <<1. Alternatively, if the cavity is small in length and the speed of the particle (c in the case of light) is large, one could approximate the situation by d/dt partial d(t)= constant d(t) and then obtain exp(-g t), having the photons no longer in deterministic motion.
According to theories of classical physics, 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 transition coefficient. Correct calculation of this coefficient is particularly important because extremely sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient was generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In the previous papers, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results were applied to calculate the half-life values of the nuclei that emit alpha particle. In this article, the half-life calculation formulas obtained in previous studies were made simpler. It has been shown that it is applicable to all radioactive decays (alpha, beta, gamma etc.). The half-life values obtained from our calculations have been compared with the experimental values, and it has been found that the new half-life values are exactly consistent with the experimental values.
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.