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arXiv:1009.2593

Remarks on the Sequential Products

Weihua Liu, Zhaoqi Wu, Junde Wu

Published Sep 14, 2010• 1 citation
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Abstract

In this paper, we show that those sequential products which were proposed by Liu and Shen and Wu in [J. Phys. A: Math. Theor. {\bf 42}, 185206 (2009), J. Phys. A: Math. Theor. {\bf 42}, 345203 (2009)] are just unitary equivalent to the sequential product $A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$.

DOI
10.48550/arxiv.1009.2593

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Approximate analytical solutions of bound states for the double ring-shaped Hulthn potential

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摘要 构造了双环形Hulthn势, 用指数函数近似表示任意分波的离心项, 运用函数分析法讨论双环型Hulthn势Schrdinger方程的束缚态解. 归一化的角向波函数和径向波函数用超几何多项式表示, 给出了束缚态能谱, 体系的束缚态的能谱方程和波函数与量子数和势参数有关. 中心势场和单环形势场角向波函数及 Hulthn势束缚态能谱是本文双环形Hulthn势的特例. 关键词: 双环形Hulthn势 / 任意分波 / 近似解析解 / 束缚态 Abstract The double ring-shaped Hulthn potential is proposed in this paper. By using the analytical method of function, the approximate analytical solutions of bound state solutions of Schrdinger equation for the double ring-shaped Hulthn potential are presented within the framework of an exponential approximation of the centrifugal potential for arbitrary l-states. The normalized angular and radial wave function expressed in terms of hypergeometric polynomials are presented. The energy spectrum equations are obtained. The wave function and the energy spectrum equations of the system are related to three quantum numbers and parameters of double ring-shaped Hulthn potential. The polar angular wave functions of the central potential and ring-shaped potential and the energy spectrum equations of Hulthn potential turn out to be the special cases of the double ring-shaped Hulthn potential. Keywords: double ring-shaped Hulthn potential / arbitrary l-wave / approximate analytical solution / bound states 作者及机构信息 陆法林, 陈昌远, 尤源 1. 盐城师范学院物理科学与电子技术学院, 盐城 224051 基金项目: 国家自然科学基金(批准号: 11275165)、江苏省自然科学基金(批准号: BK2010291)和盐城师范学院教授博士基金(批准号: 11YSYJB0206)资助的课题. Authors and contacts Lu Fa-Lin, Chen Chang-Yuan, You Yuan 1. School of Physics and Electronics, Yancheng Teachers University, Yancheng 224051, China Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11275165), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK2010291), and the Professor and Doctor Foundation of Yancheng Teachers University, China (Grant No. 11YSYJB0206). 参考文献 [1] Quesne C 1988 J. Phys. A: Math. Gen. 21 3093 [2] Chen C Y, Sun D S 2001 Acta Photon. Sin. 30 104 (in Chinese) [陈昌远, 孙东升2001光子学报 30 104] [3] Dong S H, Sun G H, Lozada-Cassou M 2004 Phys. Lett. A 328 299 [4] Hartmann H, Schuch D 1980 Int. J. Quantum Chem. 18 125 [5] Chen C Y, Liu C L, Sun D S 2002 Phys. Lett. A 305 341 [6] Zhou F, Wu Y, Guo J Y 2009 Commun. Theor. Phys. 52 813 [7] Chen C Y, Liu C L, Lu F L 2010 Phys. Lett. A 374 1346 [8] Hautot A 1973 J. Math. Phys. 14 1320 [9] Dutra A S, Hott M 2006 Phys. Lett. A 356 215 [10] Yasuk F, Durmus A. 2008 Phys. Scr. 77 015005 [11] Lu F L, Chen C Y 2010 Chin. Phys. B 19 100309 [12] Zhang M C, Huangfu G Q 2010 Acta Phys. Sin. 59 6819 (in Chinese) [张民仓, 皇甫国庆2010物理学报 59 6819] [13] Berkdemir C 2009 J. Math. Chem. 46 139 [14] Zhang M C, Sun G H, Dong S H 2010 Phys. Lett. A 374 704 [15] Ghoumaid A, Benamira F, Guechi L, Khiat Z 2013 Cent. Eur. J. Phys. 11 78 [16] Arda A, Sever R 2012 J. Math. Chem. 50 1484 [17] Maghsoodi E, Hassanabadi H, Zarrinkamar S 2013 Chin. Phys. B 22 030302 [18] Guo J Y, Meng J, Xu F X 2003 Chin. Phys. Lett. 20 602 [19] Zhang M C 2008 Chin. Phys. B 17 3214 [20] Chen C Y, Sun D S, Lu F L 2008 J. At. Mol. Phys. 25 393 (in Chinese) [陈昌远, 孙东升, 陆法林2008 原子与分子物理学报 25 393] [21] Qiang W C, Chen W L, Li K, Zhang H P 2009 Int. J. Mod. Phys. A 24 5523 [22] Feng J S, Liu Z, Guo J Y 2010 Chin. Phys. Lett. 27 110304 [23] Zhang M C, HuangFu G Q 2011 Chin. Phys. Lett. 28 50304 [24] Agboola D 2011 Commun. Theor. Phys. 55 972 [25] Liu S K, Liu S D 2002 Special Function (Beijing: China Meteorological Press) p8, 485, 571, 585 (in Chinese) [刘式适, 刘式达 2002 特殊函数(北京: 气象出版社) 第8, 485, 571, 585页] [26] Gradsgteyn I S, Ryzhik M 2007 Tables of Integrals, Series, and Products (7th Ed.) (New York: Academic Press) p971 [27] Chen C Y, Lu F L, Sun D S, Dong S H 2013 Chin. Phys. B 22 100302 施引文献

Acta Physica Sinica•Jan 1, 2013•3 citations
In this paper, we show that those sequential products which were proposed by Liu and Shen and Wu in [J. Phys. A: Math. Theor. {\bf 42}, 185206 (2009), J. Phys. A: Math. Theor. {\bf 42}, 345203 (2009)] are just unitary equivalent to the sequential product A∘B=A12BA12A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}A∘B=A21​BA21​.
A quantum effect is an operator AAA on a complex Hilbert space HHH that satisfies 0≤A≤I0\leq A\leq I0≤A≤I, E(H){\cal E} (H)E(H) is the set of all quantum effects on HHH. In 2001, Professor Gudder and Nagy studied the sequential product A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 of A,B∈E(H)A, B\in {\cal E}(H)A,B∈E(H). In 2005, Professor Gudder asked: Is A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 the only sequential product on E(H){\cal E} (H)E(H)? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on E(H){\cal E} (H)E(H); secondly, we present a general method for constructing sequential products on E(H){\cal E} (H)E(H); finally, we study some properties of the sequential products constructed by the method
Abstract This paper is a rebuttal of the comment by Zhdanov (2018 J. Phys. A: Math. Theor .) with regard to our paper (Russell et al 2017 J. Phys. A: Math. Theor . 50 205302). Parts of the comment either misrepresent the original claims in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or consider issues that go beyond the scope of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). We address all of these points in the rebuttal. A remaining item about the prospect of an accumulation of null sets was already considered in a prior work (Russell et al 2018 J. Phys. A: Math. Theor . 51 335103). Importantly, the comment fails to consider the vast and continually growing set of successful quantum control tests under well-defined conditions, which serves as foundational evidence for the results in Russell et al (2017 J. Phys. A: Math. Theor . 50 205302). In summary, the remarks in the comment either misconstrue the claims of Russell et al (2017 J. Phys. A: Math. Theor . 50 205302) or raise points that are irrelevant to the goal of the original work. Thus, we conclude that the primary claim of the original paper stands: almost all closed finite dimensional quantum systems are trap free given adequate control resources.
摘要 构造了双环形Hulthn势, 用指数函数近似表示任意分波的离心项, 运用函数分析法讨论双环型Hulthn势Schrdinger方程的束缚态解. 归一化的角向波函数和径向波函数用超几何多项式表示, 给出了束缚态能谱, 体系的束缚态的能谱方程和波函数与量子数和势参数有关. 中心势场和单环形势场角向波函数及 Hulthn势束缚态能谱是本文双环形Hulthn势的特例. 关键词: 双环形Hulthn势 / 任意分波 / 近似解析解 / 束缚态 Abstract The double ring-shaped Hulthn potential is proposed in this paper. By using the analytical method of function, the approximate analytical solutions of bound state solutions of Schrdinger equation for the double ring-shaped Hulthn potential are presented within the framework of an exponential approximation of the centrifugal potential for arbitrary l-states. The normalized angular and radial wave function expressed in terms of hypergeometric polynomials are presented. The energy spectrum equations are obtained. The wave function and the energy spectrum equations of the system are related to three quantum numbers and parameters of double ring-shaped Hulthn potential. The polar angular wave functions of the central potential and ring-shaped potential and the energy spectrum equations of Hulthn potential turn out to be the special cases of the double ring-shaped Hulthn potential. Keywords: double ring-shaped Hulthn potential / arbitrary l-wave / approximate analytical solution / bound states 作者及机构信息 陆法林, 陈昌远, 尤源 1. 盐城师范学院物理科学与电子技术学院, 盐城 224051 基金项目: 国家自然科学基金(批准号: 11275165)、江苏省自然科学基金(批准号: BK2010291)和盐城师范学院教授博士基金(批准号: 11YSYJB0206)资助的课题. Authors and contacts Lu Fa-Lin, Chen Chang-Yuan, You Yuan 1. School of Physics and Electronics, Yancheng Teachers University, Yancheng 224051, China Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11275165), the Natural Science Foundation of Jiangsu Province, China (Grant No. BK2010291), and the Professor and Doctor Foundation of Yancheng Teachers University, China (Grant No. 11YSYJB0206). 参考文献 [1] Quesne C 1988 J. Phys. A: Math. Gen. 21 3093 [2] Chen C Y, Sun D S 2001 Acta Photon. Sin. 30 104 (in Chinese) [陈昌远, 孙东升2001光子学报 30 104] [3] Dong S H, Sun G H, Lozada-Cassou M 2004 Phys. Lett. A 328 299 [4] Hartmann H, Schuch D 1980 Int. J. Quantum Chem. 18 125 [5] Chen C Y, Liu C L, Sun D S 2002 Phys. Lett. A 305 341 [6] Zhou F, Wu Y, Guo J Y 2009 Commun. Theor. Phys. 52 813 [7] Chen C Y, Liu C L, Lu F L 2010 Phys. Lett. A 374 1346 [8] Hautot A 1973 J. Math. Phys. 14 1320 [9] Dutra A S, Hott M 2006 Phys. Lett. A 356 215 [10] Yasuk F, Durmus A. 2008 Phys. Scr. 77 015005 [11] Lu F L, Chen C Y 2010 Chin. Phys. B 19 100309 [12] Zhang M C, Huangfu G Q 2010 Acta Phys. Sin. 59 6819 (in Chinese) [张民仓, 皇甫国庆2010物理学报 59 6819] [13] Berkdemir C 2009 J. Math. Chem. 46 139 [14] Zhang M C, Sun G H, Dong S H 2010 Phys. Lett. A 374 704 [15] Ghoumaid A, Benamira F, Guechi L, Khiat Z 2013 Cent. Eur. J. Phys. 11 78 [16] Arda A, Sever R 2012 J. Math. Chem. 50 1484 [17] Maghsoodi E, Hassanabadi H, Zarrinkamar S 2013 Chin. Phys. B 22 030302 [18] Guo J Y, Meng J, Xu F X 2003 Chin. Phys. Lett. 20 602 [19] Zhang M C 2008 Chin. Phys. B 17 3214 [20] Chen C Y, Sun D S, Lu F L 2008 J. At. Mol. Phys. 25 393 (in Chinese) [陈昌远, 孙东升, 陆法林2008 原子与分子物理学报 25 393] [21] Qiang W C, Chen W L, Li K, Zhang H P 2009 Int. J. Mod. Phys. A 24 5523 [22] Feng J S, Liu Z, Guo J Y 2010 Chin. Phys. Lett. 27 110304 [23] Zhang M C, HuangFu G Q 2011 Chin. Phys. Lett. 28 50304 [24] Agboola D 2011 Commun. Theor. Phys. 55 972 [25] Liu S K, Liu S D 2002 Special Function (Beijing: China Meteorological Press) p8, 485, 571, 585 (in Chinese) [刘式适, 刘式达 2002 特殊函数(北京: 气象出版社) 第8, 485, 571, 585页] [26] Gradsgteyn I S, Ryzhik M 2007 Tables of Integrals, Series, and Products (7th Ed.) (New York: Academic Press) p971 [27] Chen C Y, Lu F L, Sun D S, Dong S H 2013 Chin. Phys. B 22 100302 施引文献