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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 陳義裕(Yih-Yuh Chen) | |
| dc.contributor.author | Shiou-Chung Lai | en |
| dc.contributor.author | 賴修仲 | zh_TW |
| dc.date.accessioned | 2021-06-17T08:16:08Z | - |
| dc.date.available | 2021-02-22 | |
| dc.date.copyright | 2021-02-22 | |
| dc.date.issued | 2021 | |
| dc.date.submitted | 2021-01-27 | |
| dc.identifier.citation | [1] David Lindley, Uncertainty: Einstein, Heisenberg, Bohr, and the struggle for the soul of science, Anchor Books, New York, 2008, p. 59. [2] Malcolm Longair, Quantum Concepts in Physics: An Alternative Approach to the Understanding of Quantum Mechanics, Cambridge University Press, New York, 2013, p. 114. [3] Nanny Fröman and Per Olof Fröman, Physical Problems Solved by the Phase-Integral Method, Cambridge University Press, 2002. [4] David J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Pearson Prentice Hall, 2005. [5] Nanny Fröman and Per Olof Fröman, Phase-Integral Method: Allowing Nearlying Transition Points, Springer, 1996. [6] R.E. Bellman and R.E. Kalaba, Quasilinearization and Nonlinear Boundary-value Problems, Elsevier, 1965. [7] Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer, 1999. [8] James A. Murdock, Perturbations: Theory and Methods, Society for Industrial and Applied Mathematics, 1999. [9] Frederick W. Byron and Robert W. Fuller, Mathematics of Classical and Quantum Physics, Dover Publications, 1992. [10] Nanny Fröman and Per Olof Fröman, “On the application of the generalized quantal Bohr-Sommerfeld quantization condition to single-well potentials with very steep walls,” J. Math. Phys. 19 (1978) 1823-1829. [11] J. L Dunham, “The Wentzel-Brillouin-Kramers Method of Solving the Wave Equation,” Phys. Rev. 41 (1932) 713-720. [12] H. Friedrich and J. Trost, “Phase Loss in WKB Waves Due to Reflection by a Potential,” Phys. Rev. Lett. 76 (1996), 4869-4873. [13] H. Friedrich and J. Trost, “Nonintegral Maslov indices,” Phys. Rev. A 54 (1996) 1136-1145. [14] M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35 (1972) 315-397. [15] N. T. Maitra and E. J. Heller, “Semiclassical perturbation approach to quantum reflection,” Phys. Rev. A 54 (1996) 4763-4769. [16] K. Raghunathan and R. Vasudevan, “JWKB method and quasilinearisation,” J. Phys. A: Math. Gen. 20 (1987) 839-845. [17] Alexander Jurisch, “Beyond WKB quantum corrections to Hamilton-Jacobi theory,” J. Phys. A: Math. Theor. 40 (2007) 10829-10849. [18] M. Maggia, S.A. Eisa and H.E. Taha, “On higher-order averaging of time-periodic systems: reconciliation of two averaging techniques,” Nonlinear Dyn. 99 (2020) 813-836. [19] Rudolph E. Langer, “On the Connection Formulas and the Solutions of the Wave Equation,” Phys. Rev. 51 (1937) 669-676. [20] B.T. Polyak, “Newton-Kantorovich Method and Its Global Convergence,” J. Math. Sci. 133 (2006) 1513–1523. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/73994 | - |
| dc.description.abstract | 在WKB近似下的一維束縛態問題裡所得到的波耳-索末菲半整數與整數量子化條件，分別對應於兩種截然不同的邊界條件，但當實際的邊界條件界於兩者之間時，兩個量子化條件的適用性將成為問題。我們知道的是，在系統兩側的位能牆逐漸傾斜至無窮斜的過程裡，半整數量子化條件的適用性終將被整數量子化條件所取代，但在絕大多數的情況裡，我們對於其中過渡地帶的細節仍未知曉。
本論文旨在探討這種過渡發生的過程與機轉，在探索的過程裡，我們得到了WKB近似的另一種推導，並分別使用了兩種不同的近似方法來計算在過渡發生的過程中，相位在位能牆上的變化。 | zh_TW |
| dc.description.abstract | The Bohr-Sommerfeld half-integer and integer quantization conditions, obtained when the WKB approximation is applied to two types of systems with different boundary conditions, become ambiguous and subtle if neither case is an accurate description of the actual system. As a not-so-steep potential wall gradually becomes infinitely steep, it is a known fact that the half-integer rule should be eventually replaced by the integer rule, but the transition in between, in most cases, is left unknown.
This thesis aims to explore the question of how and why such transition occurs. In the process of doing so, we present an alternative derivation of the WKB approximation and apply two different methods to calculate the change in the 'phase loss' during such transition. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-17T08:16:08Z (GMT). No. of bitstreams: 1 U0001-2601202122303900.pdf: 2387610 bytes, checksum: 7d0a2e29ec98291f5d181a111def2567 (MD5) Previous issue date: 2021 | en |
| dc.description.tableofcontents | 口試委員會審定書 iii 誌謝 v 摘要 vii Abstract ix 1 Introduction 1 2 WKB Approximation and Its Shortcomings 5 2.1 WKB Approximation: A Review . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Non-uniformity Across the Transition . . . . . . . . . . . . . . . . . . . 8 2.3 An Alternative Derivation: Method of Averaging . . . . . . . . . . . . . 10 3 Beyond the WKB Approximation 19 3.1 Quasi-linearization Method . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Green’s Function Method Based on Uniform Approximations . . . . . . . 24 4 Conclusion 29 Bibliography 31 | |
| dc.language.iso | en | |
| dc.subject | 波耳索末菲量子化條件 | zh_TW |
| dc.subject | WKB近似 | zh_TW |
| dc.subject | Bohr-Sommerfeld quantization condition | en |
| dc.subject | WKB approximation | en |
| dc.title | 論波耳-索末菲半整數與整數量子化條件之過渡 | zh_TW |
| dc.title | On the Transition between the Bohr-Sommerfeld Half-Integer and Integer Quantization Conditions | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 109-1 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 高賢忠(Hsien-Chung Kao),魏金明(Ching-Ming Wei) | |
| dc.subject.keyword | 波耳索末菲量子化條件,WKB近似, | zh_TW |
| dc.subject.keyword | Bohr-Sommerfeld quantization condition,WKB approximation, | en |
| dc.relation.page | 32 | |
| dc.identifier.doi | 10.6342/NTU202100196 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2021-01-27 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 物理學研究所 | zh_TW |
| 顯示於系所單位： | 物理學系 | |
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