Abstract Quantum tunneling, wherein a quanject has a non-zero probability of tunneling into and then exiting a barrier of finite width and height, is the subject of Chapter 13. The description for the one-dimensional case is extended to the barrier being inverted, which forms an attractive potential well. The first application of this analysis is to the emission of alpha particles from the decay of radioactive nuclei, where the alpha-nucleus attraction is modeled by a potential well and the barrier is the repulsive Coulomb potential. Excellent results are obtained. Ditto for the similar analysis of proton burning in stars and yet a different analysis that explains tunneling through a Josephson junction, the connector between two superconductors. The final application is to the scanning tunneling microscope, a device that allows the microscopic surfaces of solids to be mapped via electrons from the surface molecules tunneling into the tip of the STM probe.
Abstract Quantum tunneling, wherein a quanject has a non-zero probability of tunneling into and then exiting a barrier of finite width and height, is the subject of Chapter 13. The description for the one-dimensional case is extended to the barrier being inverted, which forms an attractive potential well. The first application of this analysis is to the emission of alpha particles from the decay of radioactive nuclei, where the alpha-nucleus attraction is modeled by a potential well and the barrier is the repulsive Coulomb potential. Excellent results are obtained. Ditto for the similar analysis of proton burning in stars and yet a different analysis that explains tunneling through a Josephson junction, the connector between two superconductors. The final application is to the scanning tunneling microscope, a device that allows the microscopic surfaces of solids to be mapped via electrons from the surface molecules tunneling into the tip of the STM probe.
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.
The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.
In this dissertation, I have studied four theoretical problems in quantum tunneling, quantum computing, and high-temperature superconductivity. \n I have developed a generally-useful numerical tool for analyzing impurity-induced resonant-state images observed with scanning tunneling microscope (STM) in high temperature superconductors. The integrated tunneling intensities on all predominant sites have been estimated. The results can be used to test the predictions of any tight-binding model calculation. \n I have numerically simulated two-dimensional time-dependent tunneling of a Gaussian wave packet through a barrier, which contains charged ions. We have found that a negative ion in the barrier directly below the tunneling tip can deflect the tunneling electrons and drastically reduce the probability for them to reach the point in the target plane directly below the tunneling tip. \n I have studied an infinite family of sure-success quantum algorithms, which are introduced by C.-R. Hu [Phys. Rev. A {\\bf 66}, 042301 (2002)], for solving a generalized Grover search problem. Rigorous proofs are found for several conjectures made by Hu and explicit equations are obtained for finding the values of two phase parameters which make the algorithms sure success. \n Using self-consistent Hartree-Fock theory, I have studied an extended Hubbard model which includes quasi-long-range Coulomb interaction between the holes (characterized by parameter V). I have found that for sufficiently large V/t, doubly-charged-antiphase-island do become energetically favored localized objects in this system for moderate values of U/t, thus supporting a recent conjecture by C.-R. Hu [Int. J. Mod. Phys. B {\\bf 17}, 3284 (2003)].
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.
Alpha decay Abstract Alpha decay is one of the foremost and most well- studied forms of radioactive decay, involving the emigration of a nascence flyspeck — two protons and two neutrons bound together — from an unstable infinitesimal nexus. Classical drugs, with its energy- grounded rules, suggests that such a flyspeck should be confined within the nexus indefinitely, unfit to overcome the immense implicit hedge posed by nuclear and electrostatic forces. Yet, in reality, nascence patches escape routinely, with remarkably harmonious powers and predictable half- lives This incongruity remained undetermined until the emergence of amount mechanics in the early 20th century, which introduced the conception of amount tunneling — a miracle where patches have a finite probability of" tunneling" through walls they warrant the energy to surmount. This paper takes a detailed look at the drugs of amount tunneling and its part in enabling nascence decay. We explore the literal development of the proposition, the fine frame behind tunneling chances, and the deep perceptivity this conception provides into nuclear geste. We also examine real- world exemplifications, similar as uranium- 238, and highlight broader operations of tunneling in fields ranging from nuclear drug to amount computing.
This chapter explores several cases of one-dimensional quantum motion, emphasizing the physical insights and analytical results. The particle in a rectangular potential well is examined, including the derivation of its quantized energy spectrum and normalization constant, particularly for the case of an infinitely deep well. Comparisons are drawn between the quantum and classical descriptions of motion. The analysis extends to a rectangular potential barrier, where reflection and transmission coefficients are calculated, and the phenomenon of quantum tunneling is explored. The tunnel effect paradox is addressed, along with its applications to α-decay and the cold emission of electrons from metals. Finally, the linear harmonic oscillator is treated in detail, including its energy eigenvalues, eigenstates, parity properties, and zero-point energy — the latter discussed as a consequence of the uncertainty principle and the wave-like behavior of microscopic particles. Metadata/Keywords: one-dimensional motion, rectangular potential well, α-decay, cold emission, quantum tunneling, linear harmonic oscillator, zero-point energy, parity, uncertainty principle, energy eigenstates