Course Name: Quantum Mechanics ( 量子力學 ) Part II Applications Report : C. K. Lee

1. Course Name:Quantum Mechanics (量子力學) Part II Applications Report :C. K. Lee Major References: 1. J. G. David, 2005, “Introduction to Quantum Mechanics”, Pearson Prentice Hall. 2. P. W. Atkins, 1998, “Physical Chemistry”, Oxford University Press. Course Outline: 國立中正大學 • Scattering ☺☺ • The adiabatic approximation ☺☺ • Time-dependent perturbation theory ☺☺ • The WKB approximation ☺☺ • The variational principle ☺

2. Classical scattering theory 2-D

3. 3-D

4. Example: Hard-sphere scattering

5. Rutherford scattering In order to find out more about the structure of the atom. Ernest Rutherford fired alpha particles at a thin sheet of gold foil. Alpha particles have a positive charge and so does the nucleus of a gold atom, hence they repel each other and the alpha particles are scattered when they pass close by. The alpha particles produce a tiny spark of light when they hit a fluorescent screen which can then be detected. The apparatus Rutherford used looked like this: References by [http://www.waowen.screaming.net/revision/nuclear/rsanim.htm]

6. Quantum scattering theory

7. 3-D

8. Partial wave analysis I. II. combine

11. Partial wave amplitudes

12. Phase shifts I. II.

14. Adiabatic Processes

20. The adiabatic approximation

21. Geometric phase

22. Quantum Interference

24. The Aharonov-Bohn Effect In 1959 Y. Aharanov and D. Bohm demonstrated that the vector potential had more physical significance that had been previously thought. They have us imagine sending electrons past a long, tightly wound solenoid that has been running for a long time.

25. Other Applications Junichiro Kono及其在萊斯大學及佛羅里達州州立大學的研究伙伴將單璧式半導體碳奈米管的溶液置於45 Tesla的強磁場中，並觀察其吸收與發射光譜，結果發現其導電帶與共價帶之間的能隙隨磁場的增加而變窄。 References: J. Kono et at. “Optical Signatures of the Aharonov-Bohm Phase in Single-Walled Carbon Nanotubes,” Science 21 May 2004:Vol. 304. no. 5674, pp. 1129 – 1131 伊利諾州大學香檳校區的Alexey Bezryadin等人也發現，原本沒有能隙的金屬型複壁式碳奈米管，在外加磁場下能隙會逐漸打開變寬，碳管因而呈半導體性；然而持續升高磁場，能隙卻又降為零而回歸金屬性。 References: A. Bezryadin et al. “h/e Magnetic Flux Modulation of the Energy Gap in Nanotube Quantum Dots ,“ Science 21 May 2004:Vol. 304. no. 5674, pp. 1132 - 1134 以上這些效應以往從未在奈米管上被觀測到，但它們都符合理論預測，而且由此突顯了的量子力學A-B效應的重要性。A-B效應雖然已在許多系統(包括碳奈米管)中出現過，但這卻是首度發現它能影響固體的能帶結構。

26. Time dependent part Total Hamiltonian Time independent part Time-dependent perturbation theory

27. Ex: Two-level systems

30. absorption and Emission of radiation atom photon

31. Ex: Laser (light amplification by stimulated emission of radiation)

33. Spontaneous Emission Einstein’s A & B coefficient Spontaneous Emission Stimulated Emission Absorption

34. Blackbody spectrum, giving the energy per unit volume, per unit frequency, for an electromagnetic field in equilibrium at temperature T.

35. Selection Rules

37. The WKB approximation

38. Classical region

39. Example for Classical region

40. Nonclassical (tunneling) region

41. Gamow’s theory of alpha decay An alpha particle contain Two protons & neutrons

43. The connection formulas

44. Patching wave function

47. Connection formulas

48. The variational principle The variational principle will get you an upper bound for Egs, which is sometimes all you need, and often, if you’re clever about it, very close to the exact value. Pick any normalized function whatsoever; I claim that Solving procedures: 1) Suggest trial wave function 2) Determine a coefficient by normalization 3) Obtain the expectation values of Hby inner-product notation 4) Differential of ＜H＞ to obtain another coefficient

49. Example 1

50. The harmonic oscillator Refer to Part I Theory