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Modern Physics Letters B

PERTURBATION APPROACH FOR QUANTUM TRANSMISSION

C. Julian Chen

Published Jan 20, 1991• 10 citations
Publisher

Abstract

In scanning tunneling microscopy (STM), the potential barriers are often close to or lower than the Fermi level, where both semiclassical tunneling theory and Bardeen’s tunneling theory are no longer valid. We present a new perturbation approach for general quantum transmission phenomena. For thick, positive potential barriers, it reduces to Bardeen’s tunneling theory. For potential barriers lower than the energy level, it is valid as long as the barrier thickness is smaller than one half of the de Broglie wavelength, a condition which is always satisfied in STM. An immediate application is shown.

DOI
10.1142/s0217984991000149

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In scanning tunneling microscopy (STM), the potential barriers are often close to or lower than the Fermi level, where both semiclassical tunneling theory and Bardeen’s tunneling theory are no longer valid. We present a new perturbation approach for general quantum transmission phenomena. For thick, positive potential barriers, it reduces to Bardeen’s tunneling theory. For potential barriers lower than the energy level, it is valid as long as the barrier thickness is smaller than one half of the de Broglie wavelength, a condition which is always satisfied in STM. An immediate application is shown.
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.
A simple way to derive Bardeen’s tunneling theory is introduced. With a low rate of tunneling of, for example, electrons from one electrode to another through a wide vacuum gap, the appearance of the second electrode can be regarded as a perturbation. Then Bardeen’s formula for tunneling follows exactly from Fermi’s golden rule, supposing that this perturbation appears suddenly at some instant of time. Bardeen’s theory is illustrated by assuming a smoothly variable barrier shape, including the rectangular one as a limiting case. This exactly solvable example clarifies the sensitivity of the tunneling probability to the barrier shape and the energy of the tunneling particle.
Parametrized Relativistic Quantum Theory (PRQT) extends conventional relativistic quantum mechanics by introducing an invariant evolution parameter (s) and a broader symmetry group. This paper presents the axiomatic and probabilistic foundations of PRQT. The resulting Stueckelberg equation is applied to the study of barrier penetration, with explicit calculations for scanning tunneling microscopy (STM). The STM application demonstrates high sensitivity to barrier width and quantitative agreement between exact and thick-barrier approximations.
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.
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.
The invention of scanning tunneling microscopy (STM) 30 years ago opened up a visual window to the nano-world and sparked off a bunch of new methods for investigating and controlling matter and its transformations at the atomic and molecular level.1 However, an adequate theoretical understanding of the method is demanding; STM images can be considered quantum theory condensed into a pictorial representation. A hands-on model is presented for demonstrating the imaging principles in introductory teaching. It uses sound waves and computer visualization to create mappings of acoustic resonators. The macroscopic simile is made possible by quantum-classical analogies between matter and sound waves. Grounding STM in acoustic experience may help to make the underlying quantum concepts such as tunneling less abstract to students.