課程名稱 |
數學物理專題 Special Topics in Mathematical Physics |
開課學期 |
105-1 |
授課對象 |
理學院 物理學研究所 |
授課教師 |
陳義裕 |
課號 |
Phys8133 |
課程識別碼 |
222 D1990 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期一2,3,4(9:10~12:10) |
上課地點 |
新物112 |
備註 |
總人數上限:60人 外系人數限制:5人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1051Phys8133_MathTop |
課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
課程大綱
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課程概述 |
This course introduces to the students some of the frequently used perturbation techniques and asymptotic analysis in physics. It is expected that the students taking this course are well-versed in materials treated in our undergraduate courses of Applied Math I-IV. Familiarity with classical mechanics and quantum mechanics is also assumed. |
課程目標 |
Outline of Contents:
(The following is only a rough outline. An * indicates a topic which might be omitted when time does not permit.)
1. Perturbation on roots of polynomials- a warm-up:
The quadratic equation revisited
Regular perturbation theory, the way not to go!
Iteration, a typically much faster converging scheme
2. Perturbation of eigenvalue problems:
Regular (Rayleigh-Schrodinger) perturbation and the solubility condition
A complex representation
Iteration, again
Degeneracy
Divergence of the perturbation series and level crossing
Rayleigh-Ritz method
3. Multiple-scale analysis:
Resonance and secular behavior
Two-timing
Method of averaging
Action-angle variables
Adiabatic invariant of classical mechanics
Periodic perturbation and Floquet theory
Mathieu’s equation and its solutions
4. Boundary layers
A model equation exhibiting boundary layers
Examples from fluid mechanics
5. Boundary perturbations:
Green function representation
Perturbing the shape of the boundary
Perturbation of the boundary conditions
6. Diffraction of scalar waves:
Green function representation
Kirchhoff approximation, Fraunhoffer and Fresnel diffraction
Sommerfeld and Rabinowicz representation
7. Asymptotic expansion of integrals:
Integral representations. Why?
Integration by parts
Laplace’s method and Watson’s lemma
Method of stationary phase
Method of steepest descent
Stokes phenomenon
Asymptotic evaluation of sums
8. Semiclassical approximation:
Hamilton-Jacobi theory
The path integral formulation
Short wave approximation
The trace formula
JWKB approximation
9. Fourier analysis*:
Gibbs phenomenon and Pinsky Phenomenon
Relevance to wave optics
10. KAM theory*:
The small-divisor problem
Continued fractions, and their applications
Perturbations do not necessarily destroy all the good stuff
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課程要求 |
Grading Policy:
1. Homework: 60%
2. One final exam: 40%
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
None, officially. |
參考書目 |
References:
1. Carl M. Bender, Steven A. Orszag , Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer (1999). NTU campus access:
http://link.springer.com/book/10.1007%2F978-1-4757-3069-2
2. R.S. Johnson, Singular Perturbation Theory, Springer (2005).
NTU campus access:
http://link.springer.com/book/10.1007/b100957
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評量方式 (僅供參考) |
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