By Zhuangqi Cao, Cheng Yin
This ebook examines the movement of microscopic debris in one-dimensional, arbitrary-shaped potentials in keeping with the analogy among Quantum Mechanics and Electromagnetism. Covers scattered sub-waves and new effects utilizing the Analytical move Matrix technique.
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Additional resources for Advances in One-Dimensional Wave Mechanics: Towards A Unified Classical View
And finally ql. Similarly, according to Eq. 20) and noting that qD ¼ ql + m + n + 1, one can obtain ql + m + n, ql + m + n À 1, . . and finally ql + m + 1. On the limit of l, m, n ! ∞ (h ! 0), ql and ql + m + 1 are in fact the attenuation coefficient at the turning points of xt1 and xt2. In the next section, we will demonstrate that ql and ql + m + 1 are positive real numbers. Further simplifying Eq. 18), it is easy to obtain κlþ1 h ¼ Nπ þ arctan ql q þ arctan lþ2 κlþ1 κ lþ1 ðN ¼ 0, 1, 2, .
1 WKB Wave Function 29 i i ψ ðxÞ ¼ Að0Þ exp SðxÞ þ Bð0Þ exp À SðxÞ : h h ð3:2Þ As mentioned above, Eq. 2) is a good approximation only when the potential or the local value of wavelength λ ¼ 2πh/p(x) varies very slowly [9, 10]. How slow? For simplicity, inserting a rightward traveling plane wave into the Schro¨dinger equation, we get a differential equation of S(x): 1 dS 2 h 1 d2 S þ ¼ E À V ðxÞ; 2m dx i 2m dx2 ð3:3Þ we now expand S(x) in the power series in h and write 2 h h S ¼ S0 þ S 1 þ S2 þ Á Á Á: i i ð3:4Þ Feeding this into Eq.
Define φj ¼ arctan qj : κj ð4:23Þ Based on Eq. 2 One-Dimensional Potential of Arbitrary Shape 55 consequently, Eq. 24) becomes 2 0 13 À Á κ jþ1 κj h þ 4φjþ1 À arctan@ tan φjþ1 A5 ¼ Nπ þ φjþ1 À φj κj À Á ðN ¼ 0, 1, 2, . Þ, j ¼ l þ 1, l þ 2, . . , l þ m À 1 : ð4:26Þ When j ¼ l + m, there is κlþm h ¼ Nπ þ arctan qlþmþ1 À φlþm : κ lþm ð4:27Þ Summing up Eq. 26) for different j, and using Eq. 27), one can obtain 13 κ jþ1 4φjþ1 À arctan@ κj h þ tan φjþ1 A5 κj j¼lþ1 j¼lþ1 0 1 q lþmþ1 A À φlþ1 : ¼ Nπ þ arctan@ κ lþm lþm X 2 0 lþmÀ1 X ð4:28Þ In view of Eq.