By Zhuangqi Cao, Cheng Yin
Advances in One-Dimensional Wave Mechanics offers a finished description of the movement of microscopic debris in one-dimensional, arbitrary-shaped potentials in accordance with the analogy among Quantum Mechanics and Electromagnetism. using a deeper figuring out of the wave nature of subject, this ebook introduces the concept that of the scattered sub-waves and a chain of latest analytical effects utilizing the Analytical move Matrix (ATM) technique. This paintings could be necessary for graduate scholars majoring in physics, customarily in uncomplicated quantum conception, in addition to for tutorial researchers exploring electromagnetism, particle physics, and wave mechanics and for specialists within the box of optical waveguide and built-in optics.
Prof. Zhuangqi Cao is a Professor of Physics at Shanghai Jiao Tong collage, China.
Dr. Cheng Yin is a instructor at Jiangsu Key Laboratory of strength Transmission and Distribution gear expertise, Hohai college, China.
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Additional resources for Advances in One-Dimensional Wave Mechanics: Towards A Unified Classical View
20) shows that the two eigenvalues of the matrix λ1 and λ2 are reciprocal to each other. Generally, the two eigenvalues can be expressed as & λ1 ¼ eiκh , λ2 ¼ eÀiκh ð2:21Þ where the physics behind κ and h are determined by the specific structure. And according to Eq. 20), apparently there is 1 1 cos ðκhÞ ¼ ðm11 þ m22 Þ ¼ TrMðhÞ, 2 2 ð2:22Þ where “Tr” denotes the trace of the matrix. 22) is an important formula in studying periodic structures, which is intimately connected with the Bloch theorem.
Trost, Working with WKB waves far from the semiclassical limit [J]. Phys. Rep. 397, 359 (2004) 11. H. Friedrich, J. Trost, Phase loss in WKB waves due to reflection by a potential [J]. Phys. Rev. Lett. 76, 4869 (1996) 12. J. Trost, C. Eltschka, H. Friedrich, Quantization in molecular potentials [J]. J. Phys. B 31, 361 (1998) 13. J. Zeng, Quantum Mechanics, vol. II [M] (Science Press, Beijing, 2007) 14. H. B. approximation for oblique incidence [J]. Physica 15, 593 (1949) 15. H. B. approximation as the first term of a geometric-optical series [J].
1. The superposition of the main wave and the subwaves forms the bound state of the potential well. From Eq. 4), if the difference between the two potential heights V1 and V2 is relatively large, the subwaves are not a small quantity and should not be ignored. Using the analytical matrix, it is easy to write down the matrix equation 2 32 3 1 1 ! cos ðκ2 h2 Þ À sin ðκ1 h1 Þ À sin ðκ 2 h2 Þ cos ðκ 1 h1 Þ ψ ð0Þ 54 5 κ1 κ2 0 ¼4 ψ ð 0Þ κ1 sin ðκ 1 h1 Þ cos ðκ1 h1 Þ κ2 sin ðκ2 h2 Þ cos ðκ 2 h2 Þ ! ψ ðh1 þ h2 Þ 0 : ð4:6Þ ψ ð h1 þ h2 Þ By simple algebraic methods, it is easy to obtain the following equation: α0 ðκ1 h1 þ κ2 h2 Þ þ ΦðsÞ ¼ nπ þ arctan κ1 α3 þ arctan ðn ¼ 0, 1, 2, .
Advances in One-Dimensional Wave Mechanics: Towards A Unified Classical View by Zhuangqi Cao, Cheng Yin