Leading to a Better Understanding of the Particle-in-a-Quantum-Corral Model. Yağ, R., Calı, A., Atik, B., Bilgen, E., Karlı, B., & Fatih Danış, M. J. Chem. Educ, 96:82–88, 2019. ![Leading to a Better Understanding of the Particle-in-a-Quantum-Corral Model [link]](https://bibbase.org/img/filetypes/link.svg "link")
Paper doi abstract bibtex Understanding the concepts of quantum mechanics has always been a challenge for undergraduate students. This is especially so because many of the introductory (analytically solvable) systems and problems discussed in textbooks are seemingly abstract. Using approximate experimental demonstrations of such systems and problems have been shown to be very helpful in teaching and illustrating the basic concepts of quantum mechanics. One such system is the particle-inside-a-ring system, whose experimental demonstration was achieved 25 years ago in the form of a "quantum corral" of iron atoms arranged in a ring on a copper surface by using scanning tunneling microscopy. However, this system, which can be a very good model to demonstrate the concepts of quantum mechanics, has not been treated in depth in the literature or in quantum-chemistry textbooks from an educational point of view. Here, we examine the particle-inside-a-ring system and its experimental demonstration in order to elucidate the difference between the superposition and statistical-mixture concepts and to provide an explicit description of how experimental data can be related (fitted) to a theoretical model. Particle-in-box systems are basic problems that help undergraduate students start getting familiar with quantum mechanics. Because the Schrö dinger equation can easily be solved for these basic systems, they are a good start for the students to progress further with a better understanding. Almost all the physical-chemistry and quantum-chemistry textbooks start with a discussion of the particle-in-a-one-dimensional-box problem (also referred to as the infinite-potential well) for demonstrating the postulates of quantum mechanics and the solution to the Schrö dinger equation. 1−11 Then, higher-dimensional boxes are considered, followed by circular and spherical boxes. Although these systems are mathematically "simple" to solve, the obtained results may not be well understood by the students without a demonstration of these results on actual physical systems. To this end, commonly discussed physical systems are π electrons of conjugated linear and cyclic molecules as approximate experimental demonstrations of the particle-in-a-one-dimensional box problem and the particle-on-a-ring problem, respectively. Rotational motion of diatomic molecules, on the other hand, is discussed in the context of the particle-on-a-sphere model. Another box problem or system that is not as commonly discussed as the above-mentioned ones is the particle-inside-a-ring system. 7,12 This system may also be referred to as the particle-in-a-disk, 13 particle-in-a-circular-box, 7 or particle-in-an-infinite-circular-potential-well system 14,15 and should not be confused with the particle-on-a-ring system. Unlike the particle-on-a-ring or particle-on-a-sphere problems, however, the particle-inside-a-ring system has not been discussed in the context of an actual physical system in the textbooks, most probably because of the lack of such physical systems appropriate or important enough for the level of undergraduate students; π electrons in (circular) polycyclic aromatic compounds like corannulene can be a good approximation of the particle-inside-a-ring system, and a laboratory exercise based on corannulene has been proposed by Hammer at al. 13 To our knowledge, the only other physical system that can be utilized as an experimental demonstration of the particle-inside-a-ring system is the "quantum corral", which was constructed by Crommie et al. in IBM Research Laboratories in a breakthrough work 25 years ago. 16 In this work, Crommie et al. constructed a circular box (quantum corral) by manipulating Fe atoms on a Cu(111) surface by using scanning tunneling microscopy (STM), as shown in Figure 1. 16 The Fe atoms, arranged in a circular geometry, behave approximately as a circular box for the surface Cu electrons. Hence, the density of the Cu electrons inside the Fe ring can be modeled by using the solutions of the Schrö dinger equation for a particle inside a ring with radius R, which are given by
@article{yag_leading_2019,
title = {Leading to a {Better} {Understanding} of the {Particle}-in-a-{Quantum}-{Corral} {Model}},
volume = {96},
url = {https://pubs.acs.org/sharingguidelines},
doi = {10.1021/acs.jchemed.8b00622},
abstract = {Understanding the concepts of quantum mechanics has always been a challenge for undergraduate students. This is especially so because many of the introductory (analytically solvable) systems and problems discussed in textbooks are seemingly abstract. Using approximate experimental demonstrations of such systems and problems have been shown to be very helpful in teaching and illustrating the basic concepts of quantum mechanics. One such system is the particle-inside-a-ring system, whose experimental demonstration was achieved 25 years ago in the form of a "quantum corral" of iron atoms arranged in a ring on a copper surface by using scanning tunneling microscopy. However, this system, which can be a very good model to demonstrate the concepts of quantum mechanics, has not been treated in depth in the literature or in quantum-chemistry textbooks from an educational point of view. Here, we examine the particle-inside-a-ring system and its experimental demonstration in order to elucidate the difference between the superposition and statistical-mixture concepts and to provide an explicit description of how experimental data can be related (fitted) to a theoretical model. Particle-in-box systems are basic problems that help undergraduate students start getting familiar with quantum mechanics. Because the Schrö dinger equation can easily be solved for these basic systems, they are a good start for the students to progress further with a better understanding. Almost all the physical-chemistry and quantum-chemistry textbooks start with a discussion of the particle-in-a-one-dimensional-box problem (also referred to as the infinite-potential well) for demonstrating the postulates of quantum mechanics and the solution to the Schrö dinger equation. 1−11 Then, higher-dimensional boxes are considered, followed by circular and spherical boxes. Although these systems are mathematically "simple" to solve, the obtained results may not be well understood by the students without a demonstration of these results on actual physical systems. To this end, commonly discussed physical systems are π electrons of conjugated linear and cyclic molecules as approximate experimental demonstrations of the particle-in-a-one-dimensional box problem and the particle-on-a-ring problem, respectively. Rotational motion of diatomic molecules, on the other hand, is discussed in the context of the particle-on-a-sphere model. Another box problem or system that is not as commonly discussed as the above-mentioned ones is the particle-inside-a-ring system. 7,12 This system may also be referred to as the particle-in-a-disk, 13 particle-in-a-circular-box, 7 or particle-in-an-infinite-circular-potential-well system 14,15 and should not be confused with the particle-on-a-ring system. Unlike the particle-on-a-ring or particle-on-a-sphere problems, however, the particle-inside-a-ring system has not been discussed in the context of an actual physical system in the textbooks, most probably because of the lack of such physical systems appropriate or important enough for the level of undergraduate students; π electrons in (circular) polycyclic aromatic compounds like corannulene can be a good approximation of the particle-inside-a-ring system, and a laboratory exercise based on corannulene has been proposed by Hammer at al. 13 To our knowledge, the only other physical system that can be utilized as an experimental demonstration of the particle-inside-a-ring system is the "quantum corral", which was constructed by Crommie et al. in IBM Research Laboratories in a breakthrough work 25 years ago. 16 In this work, Crommie et al. constructed a circular box (quantum corral) by manipulating Fe atoms on a Cu(111) surface by using scanning tunneling microscopy (STM), as shown in Figure 1. 16 The Fe atoms, arranged in a circular geometry, behave approximately as a circular box for the surface Cu electrons. Hence, the density of the Cu electrons inside the Fe ring can be modeled by using the solutions of the Schrö dinger equation for a particle inside a ring with radius R, which are given by},
journal = {J. Chem. Educ},
author = {Yağ, Rahime and Calı, Ağ and Atik, Bahar and Bilgen, Ecenaz and Karlı, Berfu and Fatih Danış, Mehmet},
year = {2019},
keywords = {Chemometrics ■ INTRODUCTION, Computational Chemistry, Computer-Based Learning, Mathematics/Symbolic Mathematics, Physical Chemistry, Problem Solving/Decision Making, Quantum Chemistry, Statistical Mechanics, Theoretical Chemistry, Upper-Division Undergraduate},
pages = {82--88},
}
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One such system is the particle-inside-a-ring system, whose experimental demonstration was achieved 25 years ago in the form of a \"quantum corral\" of iron atoms arranged in a ring on a copper surface by using scanning tunneling microscopy. However, this system, which can be a very good model to demonstrate the concepts of quantum mechanics, has not been treated in depth in the literature or in quantum-chemistry textbooks from an educational point of view. Here, we examine the particle-inside-a-ring system and its experimental demonstration in order to elucidate the difference between the superposition and statistical-mixture concepts and to provide an explicit description of how experimental data can be related (fitted) to a theoretical model. Particle-in-box systems are basic problems that help undergraduate students start getting familiar with quantum mechanics. Because the Schrö dinger equation can easily be solved for these basic systems, they are a good start for the students to progress further with a better understanding. Almost all the physical-chemistry and quantum-chemistry textbooks start with a discussion of the particle-in-a-one-dimensional-box problem (also referred to as the infinite-potential well) for demonstrating the postulates of quantum mechanics and the solution to the Schrö dinger equation. 1−11 Then, higher-dimensional boxes are considered, followed by circular and spherical boxes. Although these systems are mathematically \"simple\" to solve, the obtained results may not be well understood by the students without a demonstration of these results on actual physical systems. To this end, commonly discussed physical systems are π electrons of conjugated linear and cyclic molecules as approximate experimental demonstrations of the particle-in-a-one-dimensional box problem and the particle-on-a-ring problem, respectively. Rotational motion of diatomic molecules, on the other hand, is discussed in the context of the particle-on-a-sphere model. Another box problem or system that is not as commonly discussed as the above-mentioned ones is the particle-inside-a-ring system. 7,12 This system may also be referred to as the particle-in-a-disk, 13 particle-in-a-circular-box, 7 or particle-in-an-infinite-circular-potential-well system 14,15 and should not be confused with the particle-on-a-ring system. Unlike the particle-on-a-ring or particle-on-a-sphere problems, however, the particle-inside-a-ring system has not been discussed in the context of an actual physical system in the textbooks, most probably because of the lack of such physical systems appropriate or important enough for the level of undergraduate students; π electrons in (circular) polycyclic aromatic compounds like corannulene can be a good approximation of the particle-inside-a-ring system, and a laboratory exercise based on corannulene has been proposed by Hammer at al. 13 To our knowledge, the only other physical system that can be utilized as an experimental demonstration of the particle-inside-a-ring system is the \"quantum corral\", which was constructed by Crommie et al. in IBM Research Laboratories in a breakthrough work 25 years ago. 16 In this work, Crommie et al. constructed a circular box (quantum corral) by manipulating Fe atoms on a Cu(111) surface by using scanning tunneling microscopy (STM), as shown in Figure 1. 16 The Fe atoms, arranged in a circular geometry, behave approximately as a circular box for the surface Cu electrons. 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