The invariant eigen-operator method in matrix form and the eigenfrequency of several mesoscopic circuits

Wu Ze; Fan Hong-Yi

doi:10.7498/aps.68.20190651

Abstract

The Invariant Eigen-operator (IEO) method is widely used in solving the eigenfrequencies of the coulped quantum mesoscopic circuits. The previous IEO method is complicated but stylized, we always wasted much time in this boring processes. Here we extended the IEO method to the matrix form based on Lagrangian of the complex mesoscopic circuits, and express the ideas and processes of the previous calculations of the IEO method in a very simple matrix form. The mathematical methods we used is the indicator representation of the matrix, and we got a very simple and convenient matrix form of the IEO method. This form has important significance for the calculation of large-scale complex multi-loop mesoscopic circuits. Moreover, the matrix form of the IEO method is very friendly to the programming implementation of the complex quantum mesoscopic L−C circuits, it is probably a most optimal algorithm for calculating the eigenfrequencies of the quantum mesoscopic L−C circuits. In addition, with some help of computer programs, we used this method to calculate the eigenfrequencies of three L−C mesoscopic circuits, including two cases with and without mutual inductance. We revealed some relevant properties of these circuits by calculating results, indicating that the eigenfrequency is only related to the element properties of the mesoscopic circuit itself. Finally, we found that this method can also be used in other areas like atom-light coupling systems and solid state physics.

Keywords:

Authors and contacts

Wu Ze¹,
Fan Hong-Yi²

1.
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
2.
Department of Materials Science and Engineering, University of Science and Technology of China, Hefei 230026, China

Corresponding author: Fan Hong-Yi, fhym@ustc.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11775208)

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References

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图 1 两回路$ L\text{-}C$介观电路

Figure 1. Two-loop $L\text{-}C$ mesoscopic circuit

DownLoad: Full-Size Img PowerPoint

图 2 无互感的$ L\text-C$介观电路

Figure 2. $ L\text{-}C$ mesoscopic circuit without mutual inductance

DownLoad: Full-Size Img PowerPoint

图 3 带互感的$ L\text-C$介观电路

Figure 3. $ L\text-C$ mesoscopic circuit with mutual inductance 2m.

DownLoad: Full-Size Img PowerPoint

map

[1]	Fan H Y, Li C 2004 Phys. Lett. A 321 75 Google Scholar
[2]	任益充, 范洪义 2013 62 156301 Google Scholar Ren Y C, Fan H Y 2013 Acta Phys. Sin. 62 156301 Google Scholar
[3]	范洪义, 吴昊, 袁洪春 2011 量子力学的不变本征算符方法 (上海: 上海交通大学出版社) 第175−178页 Fan H Y, Wu H, Yuan H C 2011 Invariant Eigen-Operator Method in Quantum Mechanics (Shanghai: Shanghai Jiao Tong University Press) pp175−178 (in Chinese)
[4]	Fan H Y, Wu H, Xu X F 2005 Int. J. Mod. Phys. B 19 4073 Google Scholar
[5]	Fan H Y, Tang X B 2007 Commun. Theor. Phys. 47 865 Google Scholar
[6]	Ren G, Fan H Y 2009 Int. J. Phys. 48 2016 Google Scholar
[7]	Fan H Y, Wang T T 2007 Int. J. Mod. Phys. B 21 1961 Google Scholar
[8]	Song T Q, Fan H Y 2006 Mod. Phys. Lett. A 21 451 Google Scholar
[9]	Liu Y M 2009 Int. J. Theor. Phys. 48 2372 Google Scholar
[10]	Fan H Y, Tang X B 2006 Commun. Theor. Phys. 46 603 Google Scholar
[11]	Fan H Y, Tang X B, Hu H P 2008 Commun. Theor. Phys. 50 674 Google Scholar
[12]	Fan H Y, Wu H 2007 Mod. Phys. Lett. B 21 1751 Google Scholar
[13]	Fan H Y, Wu H 2008 Commun. Theor. Phys. 49 50 Google Scholar
[14]	Tang X B, Fan H Y 2010 Int. J. Theor. Phys. 49 877 Google Scholar
[15]	Fan H Y, Wu H 2009 Int. J. Mod. Phys. B 23 234
[16]	Fan H Y, Wang T T, Yang Y L 2006 Int. J. Mod. Phys. B 20 5417 Google Scholar
[17]	Song T Q, Zhu J Y, Fan H Y 2003 Commun. Theor. Phys. 39 91 Google Scholar
[18]	Fan H Y, Tang X B 2006 Commun. Theor. Phys. 45 1003 Google Scholar
[19]	Fan H Y, Gui W J, Gui J Z 2007 Int. J. Mod. Phys. B 21 737 Google Scholar
[20]	Yu T X, Fan H Y 2010 Commun. Theor. Phys. 53 257 Google Scholar
[21]	Meng X G, Wang J S, Zhai Y, Fan H Y 2008 Chin. Phys. Lett. 25 1205 Google Scholar
[22]	张科, 范承玉, 范洪义 2018 67 170301 Google Scholar Zhang K, Fan C Y, Fan H Y, 2018 Acta Phys. Sin. 67 170301 Google Scholar

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Vol. 68, Issue 22 - 2019-11-20

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