无限深方势阱中分数阶量子力学核的变换
Transformation of Fractional Quantum Mechanical Kernel in the Infinite Square Well
DOI: 10.12677/MP.2020.105009, PDF,    国家自然科学基金支持
作者: 陆莹, 董建平*：南京航空航天大学理学院，江苏 南京
关键词: 分数阶量子力学核；积分变换；Fox’s H函数；Fractional Quantum Mechanical Kernel； Integral Transform； Fox’s H-Function
摘要: 分数阶量子力学核是一种波函数，它可以描述分数阶量子系统的演化过程。本文研究无限深方势阱中自由粒子量子力学核的Laplace变换、能量–时间变换和动量表示。我们首先借助无限深方势阱中自由粒子量子力学核的Fox’s H函数表示，得到量子力学核的Laplace变换。然后利用量子力学核的路径积分形式，计算出它的能量–时间变换和动量表示。量子力学核的变换可以简化分数阶量子力学中的计算结果，从而更好地研究其性质。
Abstract: Fractional quantum mechanics kernel is a kind of wave function, which can describe the evolution process of fractional quantum system. In this paper, we study the Laplace transformation, ener-gy-time transformation and momentum representation of free particle quantum mechanics kernel in infinite square well. Firstly, we obtain the Laplace transformation of quantum mechanical kernel by using the Fox’s H function representation of free particle quantum mechanics kernel in the infinite square well, and then use Path integral form of quantum mechanics kernel to calculate its energy-time transformation and momentum representation. The transformation of quantum me-chanics kernel can simplify the calculation results in fractional quantum mechanics, so as to study its properties better.
文章引用：陆莹, 董建平. 无限深方势阱中分数阶量子力学核的变换[J]. 现代物理, 2020, 10(5): 79-85. https://doi.org/10.12677/MP.2020.105009

参考文献

[1] Laskin, N. (2000) Fractional Quantum Mechanics and Levy Path Integrals. Physics Letters A, 268, 298-305. [Google Scholar] [CrossRef] 
[2] Laskin, N. (2000) Fractional Quantum Mechanics. Physical Re-view E, 62, 3135-3145. [Google Scholar] [CrossRef] 
[3] Laskin, N. (2000) Fractals and Quantum Mechanics. Chaos: An Inter-disciplinary Journal of Nonlinear Science, 10, 780-790. [Google Scholar] [CrossRef] [PubMed]
[4] Laskin, N. (2002) Fractional Schrӧdinger Equation. Physical Review E, 66, Article ID: 056108. [Google Scholar] [CrossRef] 
[5] Laskin, N. (2007) Lévy Flights over Quantum Paths. Communica-tions in Nonlinear Science and Numerical Simulation, 12, 2-18. [Google Scholar] [CrossRef] 
[6] Lévy, P. (1938) Théorie De L’Addition Des Variables Aleatoires. Bulletin of the American Mathematical Society, 44, 19-20. [Google Scholar] [CrossRef] 
[7] Laskin, N. (2018) Fractional Quantum Mechanics. World Scientific Publishing Co. Pte. Ltd., Singapore. [Google Scholar] [CrossRef] 
[8] Laskin, N. (2017) Time Fractional Quantum Mechanics. Chaos, Solitons and Fractals, 102, 6-28. [Google Scholar] [CrossRef] 
[9] Dong, J.P. and Xu, M.Y. (2007) Some Solutions to the Space Frac-tional Schrӧdinger Equation Using Momentum Representation Method. Journal of Mathematical Physics, 48, Article ID: 072105. [Google Scholar] [CrossRef] 
[10] Dong, J.P. (2014) Scattering Problems in the Fractional Quantum Mechanics Governed by the 2D Space-Fractional Schrödinger Equation. Journal of Mathematical Physics, 55, Article ID: 032102. [Google Scholar] [CrossRef] 
[11] Larkin, A.S., Filinov, V.S. and Fortov, V.E. (2016) Path Integral Representation of the Wigner Function in Canonical Ensemble. Contributions to Plasma Physics, 56, 187-196. [Google Scholar] [CrossRef] 
[12] Dong, J.P. and Geng, H. (2018) Levy Path Integrals of Particle on Circle and Some Applications. Journal of Mathematical Physics, 59, Article ID: 112103. [Google Scholar] [CrossRef] 
[13] Dong, J.P. (2013) Lévy Path Integral Approach to the Solution of the Fractional Schrӧdinger Equation with Infinite Square Well.
[14] Mathai, A.M. and Saxena, R.K. (1978) The H-Function with Applications in Statistics and Other Disciplines. Wiley Eastern, New York.
[15] Mathai, A.M., Saxena, R.K. and Haubold, H.J. (2010) The H-Function: Theory and Applications. Springer, New York. [Google Scholar] [CrossRef] 
[16] Fox, C. (1961) The G and H-Functions as Symmetrical Fourier Kernels. Transactions of the American Mathematical Society, 98, 395-395. [Google Scholar] [CrossRef] 
[17] Prudnikov, A.P., Brychkov, Y.A., Gould, C.G.T., et al. (1990) Integrals and Series More Special Functions. Mathematics of Computation, 44, 573.
[18] Nayga, M.M. and Es-guerra, J.P. (2016) Green’s Functions and Energy Eigenvalues for Delta-Perturbed Space-Fractional Quantum Systems. Journal of Mathematical Physics, 57, 3081-3122. [Google Scholar] [CrossRef]