課程名稱︰近代物理 課程性質︰電機系複選必修 課程教師︰吳志毅 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰2021/6/22 考試時限（分鐘）：170分鐘 試題 : 註：本次考試是線上開書考 Problem 1 (10%) The Maxwell distribution of molecular velocity is n(v)dv=4πN( m/2πkT )^(3/2)v^2exp(m v^2/ 2kT)dv Please calculate the average value of (1) 1/v and (2) v Problem 2 (10%) (a) Why can a small vibration of any potential force be approximated as a simple harmonic oscillator? (b) What are the differences between fluorescence and phosphorescence? Problem 3 (10%) In a system with 2 electrons which follows the Pauli Exclusion Principle, please explain why the total wavefunction has to be antisymmetric? Problem 4 (15%) Consider fcc, bcc, sc, and diamond structures. (a) What are the coordination numbers of fcc, bcc, sc, and diamond structure, respectively? (b) Suppose identical solid sphere are distributed through space in such a way that their centers lie on the points of each of these four structures, and spheres on neighboring points just touch, without overlapping. (This is called close-packing arrangement.) Assuming that the sphere have density of one, what is the density of a set of close-packed spheres on each of these four structures? (c) Which structure is the most dense and which one is the least dense structure? Problem 5: (10%) For a simple cubic lattice with lattice constant a, please write down the coordination of all the atoms in (a) the first nearest neighbors, (b) the second nearest neighbors, and (c) the third nearest neighbors, with respect to the atom at the origin (0, 0, 0). Problem 6 (10%) The potential energy between two atoms in a molecule can often be written as U(r) = U_0 - [ (a/r)^12 - 2(a/r)^6 ] Where U0 and a are constants. (a) Find the equilibrium interatomic separation r0 in terms of a. (b) Find the corresponding value of potential energy. Problem 7 (10%) (a) Show that the rule of Dulong-Petit follows directly from Einstein’s specific heat formula as T→∞ C_v = dU/dT = 3*N_a*k*(hf/kT)^2* exp(hf/kT) / (exp(hf/kT)-1)^2 (b) Find Cv as T→0 Problem 8 (15%) Consider a system of N particles that has only two possible energy states, E1=0 and E2=ε. The distribution function is fi = C*exp(-Ei/kT) (a) What is C for this case ? (b) Compute the average energy <E> and show that <E>→0 as T→0 and <E>→ε /2 as T→∞. (c) Calculate the heat capacity Problem 9 (10%) In the anomalous Zeeman effect, the external magnetic field is much weaker than the internal field seen by the electron as a result of its orbital motion. In the vector model the vectors L and S precess rapidly around J because of the internal field and J precesses slowly around the external field. The energy splitting is found by first calculating the component of the magnetic moment μJ in the direction of J and then finding the component of μZ in the direction of B. (a) Please calculate μJ in terms of μΒ , J, S, and L (b) Please calculate μZ in terms of μΒ , JZ, J, S, and L --

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