课程名称︰量子物理下 课程性质︰必修 课程教师︰陈义裕 开课学院：理学院 开课系所︰物理系 考试日期（年月日）︰97/6/11 考试时限（分钟）：120分钟 是否需发放奖励金：是 谢谢 （如未明确表示，则不予发放） 试题 : 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. --

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