課程名稱︰量子物理下 課程性質︰必修 課程教師︰陳義裕 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰97/6/11 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 謝謝 （如未明確表示，則不予發放） 試題 : 1.(25points)It is known that the average thermal energy for a normal mode of a crystal as the temperature T can be taken as hω ε = ────── h:Planck constant/2π exp(βhω)-1 where ω is the (angular) frequency of the normal mode, and β=1/(kbT), kb:Boltzmann constant We will consider Debye's model of specific heat for a linear chain consisted of N identical atoms. However, we assune that, corresponding to a given wavenumber k, there are three types of allowed vibrational motion of the crystal lattice : two transverse and one longitudinal. For simplicity, we assume that the dispersion relation satisfies ω=c∣k∣for all the normal modes having a wavenumber k with c signifying the wave speed. (a)(5points)Using periodic boundry conditions, please explain why it is natural that k=2πj/Na, where j is an integer ranging from -N/2 to N/2. (b)(5points)Please show that the total thermal energy of crystal is 3Nkb(T^2) Θ/T x E = ──────∫ ── dx Θ 0 exp(x)-1 (c)(5points)For T very large, please show that the specific heat per atom is approxiamtion 3kb. (d)(5points)For T very small, how does the specific heat depend on the temperature T? (e)(5points)If we generalize this model to a two-dimensional lattice, keeping the same three types of vibration and the same dispersion relation for all the normal modes, then what is the temperature dependence of the specific heat as low T? 2.(25points) (a)(10points)Please argue why the relative magnitude of the elcectronic energy, the molecular vibrational energy,and the molecular rotational energy is approximately of the ratio m m 1:√ ──:── M M where m and M are the mass of electron and the nucleus, respectively .Then please utilize this fact to give a brief account of the so-called Born-Oppenheimer approximation of the structure of molecules. (b)(8points)Consider two hypothetical homonuclear diatomic molecules A2 and B2. It is known that both the atomic nu,ber and the mass number for Nucleus A and Nucleus B are the same. The only known difference between A and B is that A is a fermion while B is a boson. Please explain why A2 and B2 molecular are not expected to have the same vibration and rotation molechlar spectra. (c)(7points)Please explain how one may use the experimental data of the dissociation energy of molecules to indirectly prove the zero-point energy of a simple harmonic oxcillator is 1/2 hω. --

※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 218.168.68.56 ※ 編輯: leehoni 來自: 218.168.68.56 (06/12 22:32) ※ 編輯: leehoni 來自: 218.168.75.246 (06/13 17:26) ※ PHONm:轉錄至看板 NTUscuba 11/28 14:12