边界对二维反谐振子在平方反比势和楔子中逗留时间的影响
PDF下载 (152)乐家怡,陈 寅,潘孝胤.边界对二维反谐振子在平方反比势和楔子中逗留时间的影响[J].宁波大学学报(理工版),2022,35(1):105-111.DOI:
LE Jiayi,CHEN Yin,PAN Xiaoyin.Boundary effects on the sojourn time of two-dimensional inverted inverse quadratic oscillator in a wedge[J].Journal of Ningbo University(Natural Science & Engineering Edition),2022,35(1):105-111.DOI:
| Title: | Boundary effects on the sojourn time of two-dimensional inverted inverse quadratic oscillator in a wedge |
| 作者: | 乐家怡, 陈 寅, 潘孝胤 |
| Author(s): | LE Jiayi, CHEN Yin, PAN Xiaoyin |
| 关键词: | 边界效应; 逗留时间; 路径积分; 楔子 |
| Keywords: | boundary effect; sojourn time; path-integral; wedge |
| 分类号: | O413.1 |
| 文献标识码: | A |
| 摘要: | 本文研究了二维反谐振子在平方反比势和外角为 ( )的楔子中的逗留时间, 利用费曼路径积分法得到了波函数和相应密度的精确表达式, 进一步利用数值积分方法计算了逗留时间与a的函数. 结果表明, 随着外角增加, 逗留时间单调递增. 在 的极限上有一个很大的跳跃, 这意味着当楔子成为奇异线时, 与没有楔子时存在着很大的差异. 换句话说, 尽管奇异线在二维空间中并不占据任何区域, 但是由于奇异线的存在, 使得逗留时间大大缩短, 从而说明了边界条件的重要性. |
| Abstract: | By employing the Feynman path-integral approach, we investigate the sojourn time of the two-dimensional inverted inverse quadratic oscillator in a wedge with external angle ( ). The exact expressions for the wave function and the corresponding density are obtained analytically, then the sojourn time as a function of a is calculated numerically. It is shown that the sojourn time monotonically increases as the external angle increases, but there is a big jump in the limit of implying that there is a huge difference between the case when the wedge becomes a singular line and the one without a wedge. In other words, although a singular line does not occupy any area in the two-dimensional space, the sojourn time can still be reduced dramatically due to the existence of the singular line, which may indicate the importance of the boundary condition. |
| 参考文献 /References: | [1] Jackson J D. Classical Electrodynamics[M]. 3rd ed. Zurich: Wiley-VCH, 1998:832. [2] Glimm J, Jaffe A. Quantum Physics—A Functional Integral Point of View[M]. Berlin: Springer-Verlag, 1987. [3] Klitzing K V, Dorda G, Pepper M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance[J]. Physical Review Letters, 1980, 45(6):494-497. [4] MacDonald A H. Quantum Hall Effect: A Perspective[M]. Dordrecht: Springer Netherlands, 1989. [5] Chakraborty T, Pietil?inen P. The Fractional Quantum Hall Effect[M]. Heidelberg: Springer-Verlag, 1988. [6] Prange R E, Girvin S M. The Quantum Hall Effect[M]. New York: Springer, 1990. [7] Chakraborty T. Quantum Dots[M]. Amsterdam: Elsevier, 1999. [8] Jacak L, Harylak P, Wojs A. Quantum Dots[M]. Heidelberg: Springer-Verlag, 2013. [9] Reimann S M, Manninen M. Electronic structure of quantum dots[J]. Reviews of Modern Physics, 2002, 74(4):1283-1342. [10] Hanson R, Kouwenhoven L P, Petta J R, et al. Spins in few-electron quantum dots[J]. Reviews of Modern Physics, 2007, 79(4):1217-1265. [11] Harrison P, Valavanis A. Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures[M]. 4th ed. Hoboben, NJ: John Wiley & Sons, 2016:505-542. [12] Novoselov K S, McCann E, Morozov S V, et al. Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene[J]. Nature Physics, 2006, 2(3):177-180. [13] Castro Neto A H, Guinea F, Peres N M R, et al. The electronic properties of graphene[J]. Reviews of Modern Physics, 2009, 81(1):109-162. [14] Kotov V N, Uchoa B, Pereira V M, et al. Electron-electron interactions in graphene: Current status and perspectives[J]. Reviews of Modern Physics, 2012, 84(3):1067-1125. [15] Birrell N D, Davies P C W. Quantum Fields in Curved Space[M]. Cambridge: Cambridge University Press, 1984. [16] Preskill J. Cosmic strings and other topological defects[J]. Science, 1996, 272(5264):966-971. [17] Grosche C. δ-Function perturbations and boundary problems by path integration[J]. Annalen der Physik, 1993, 505(6):557-589. [18] Müller-Kirsten H J W. Introduction to Quantum Mechanics: Schr?dinger Equation and Path Integral[M]. Singapore: World Scientific Publishing, 2006. [19] Feynman R P, Hibbs A R. Quantum Mechanics and Path Integrals[M]. New York: McGraw-Hill, 1965. [20] Feynman R P. Space-time approach to non-relativistic quantum mechanics[J]. Reviews of Modern Physics, 1948, 20(2):367-387. [21] Asorey M, Clemente-Gallardo J, Mu?oz-Casta?eda J M. Boundary conditions: The path integral approach[J]. Journal of Physics: Conference Series, 2007, 87:012004. [22] von Sommerfeld A. ?ber verzweigte potentiale im raum[J]. Proceedings of the London Mathematical Society, 1896, 28(1):395-429. [23] Pauli W. On asymptotic series for functions in the theory of diffraction of light[J]. Physical Review, 1938, 54(11):924-931. [24] Crandall R E. Combinatorial approach to Feynman path integration[J]. Journal of Physics A: General Physics, 1993, 26(14):3627-3648. [25] DeWitt-Morette C. Feynman path integrals: I. Linear and affine techniques; II. The Feynman-Green function[J]. Communications in Mathematical Physics, 1974, 37(1):63-81. [26] Schulman L S. Exact time-dependent Green’s function for the half-plane barrier[J]. Physical Review Letters, 1982, 49(9):599-601. [27] Wiegel F W, Michels J J. Dynamics of unwinding of a simple entanglement[J]. Journal of Physics A: General Physics, 1987, 20(14):4653-4659. [28] Cheng B K. The two-dimensional harmonic oscillator interacting with a wedge[J]. Journal of Physics A: General Physics, 1990, 23(24):5807-5814. [29] Inomata A. Remarks on the Time Transformation Technique for Path Integration[M]//Gutzwiller M C, Inomata A, Klauder J R, et al. Path Integrals from meV to MeV. Singapore: World Scientific Publishing, 1986:433-448. [30] Cheng B K, Luz M E. Exact propagator for a two-dimensional inverse quadratic oscillator interacting with a wedge[J]. Journal of Physics A: General Physics, 1992, 25(7):2033-2042. [31] Barton G. Quantum mechanics of the inverted oscillator potential[J]. Annals of Physics, 1986, 166(2):322-363. [32] Baskoutas S, Jannussis A, Mignani R. Study of the generalized parametric oscillator[J]. Physics Letters A, 1992, 164(1):17-22. [33] Baskoutas S, Jannussis A, Mignani R. Quantum tunnelling of a damped and driven, inverted harmonic oscillator[J]. Journal of Physics A: General Physics, 1993, 26(23):7137-7147. [34] Guo G J, Ren Z Z, Ju G X, et al. Quantum tunneling effect of a time-dependent inverted harmonic oscillator[J]. Journal of Physics A: Mathematical & Theoretical, 2011, 44(18):185301. [35] Guo G J, Ren Z Z, Ju G X, et al. Quantum tunneling effect of a driven inverted harmonic oscillator[J]. Journal of Physics A: Mathematical and Theoretical, 2011, 44(30):305301. [36] Guo G J, Ren Z Z, Ju G X, et al. The sojourn time of the inverted harmonic oscillator on the noncommutative plane[J]. Journal of Physics A: Mathematical and Theoretical, 2011, 44(4):425301. [37] Felder G, Frolov A, Kofman L, et al. Cosmology with negative potentials[J]. Physical Review D, 2002, 66(2):023507. [38] AbuSalbi N, Kouri D J, Shima Y, et al. Quantum mechanical study of the D+H2→HD+H reaction[J]. The Journal of Chemical Physics, 1985, 82(6):2650-2661. [39] Fahyan S E. Laser physics[D]. Malang, Indonesia: Universitas Negeri Malang, 1993. [40] Bender C M, Cooper F, Milton K A, et al. Discrete-time quantum mechanics. III. Spin systems[J]. Physical Review D, 1987, 35(10):3081-3091. [41] Wang Y J, Guo W, Pan X Y, et al. The Aharonov-Bohm effect on the sojourn time of the two-dimensional inverted harmonic oscillator under a perpendicular magnetic field[J]. Europhysics Letters, 2019, 124(6):60007. [42] Bogachek E N, Landman U. Edge states, Aharonov- Bohm oscillations, and thermodynamic and spectral properties in a two-dimensional electron gas with an antidot[J]. Physical Review B, 1995, 52(19):14067-14077. [43] Gradshteyn I S, Ryzhik I M, Jeffrey A, et al. Table of integrals, series, and products[J]. Journal of Biomechanical Engineering, 1981, 103(1):58. [44] Wang Z, Guo D. An Introduction to Special Function[M]. Beijing: Peking University Press, 2000. (in Chinese) |
| 备注/Memo: | 收稿日期: 2021－10－08. 宁波大学学报（理工版）网址: http://journallg.nbu.edu.cn/ 基金项目: 国家自然科学基金（11375090）. 第一作者: 乐家怡（1996－）, 女, 浙江慈溪人, 在读硕士研究生, 主要研究方向: 量子力学. E-mail: 592981258@qq.com *通信作者: 潘孝胤（1974－）, 男, 浙江宁海人, 研究员, 主要研究方向: 凝聚态理论. E-mail: panxiaoyin@nbu.edu.cn 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
