量子物理课堂习题.docx

量子物理课堂习题Lecture 1: 旧量子论1. 求氘原子H线n=2到 n=3的波数2. Ce的逸出功是1.9eV, 求阈值频率和波长3. 对于氢原子、一次电离的氦离子 He+和两次电离的锂粒子 Li+,分别计算它们的： a) 第一、第二波尔轨道半径及电子在这些轨道上的速度b) 电子在基态的结合能c) 第一激发态退激到基态所放光子的波长Lecture 2：波粒二象性 不确定性原理1. 已知琴弦振动的驻波条件为n2=a（n1，2，, a 为弦长）。按照“定态即驻波”的说法，束缚在长宽高分别为 a，b，c 的三维势箱中的粒子(质量为 m)的定态能量取值是多少？2. 一原子的激发态发射波长为 600nm 的光谱线，测得波长的精度为/=10-7 ,试问该原子态的寿命为多长？3. 1，3丁二烯分子长度a7，试用测不准关系估计其基电子态能级的大小（量级）Lecture 3: 波函数 薛定谔方程1. 下列哪些函数不是品优函数，说明理由：fx= x2, e-x,sinx, e-x22. 试写出下列体系的定态薛定谔方程：（a）He 原子（b）H2 分子3. 写出一个被束缚在半径为a的圆周上运动的粒子的 Schrdinger 方程，并求其解Lecture 4: 势箱模型1. (2.12) For the particle in a one-dimensional box of length 1, we could have put the coordinate origin at the center of the box. Find the wave functions and energy levels for this choice of origin.2.3. （2.5）Consider a particle with quantum number n moving in a one-dimensional box of length l. (a) Determine the probability of finding the particle in the left quarter of the box. (b) For what value of n is this probability a maximum? (c) What is the limit of this probability for n? (d) What principle is illustrated in (c)?4. （A23）试用一维势箱模型(6个电子)计算如下分子的电子光谱最大吸收波长（第一吸收峰）。Lecture 5: 谐振子1. （4.13） Use the recursion relation (4.48) to find the =3 normalized harmonic-oscillator wave function. 2. （4.18）The three-dimensional harmonic oscillator has the potential-energy function V=12kxx2+12kyy2+12kzz2, where the ks are three force constants. Find the energy eigenvalues by solving the Schrodinger equation. (b) If kx=ky=kz,find the degree of degeneracy of each of the four lowest energy levels.3. （4.22）Find the eigenvalues and eigenfunctions of H for a one-dimensional system with Vx= for x0, Vx=12kx2 for x0. Lecture 6: 算符与量子力学1. （3.39）For the particle confined to a box with dimensions a, b, and c, find the following values for the state with quantum numbers nx, ny, nz. (a) x; (b) y,z . Use symmetry considerations and the answer to part a. (c)px; (d)x2 . Is x2=x2? Is xy=xy?2. （3.24）Find the eigenfunctions of -22md2/dx2. If the eigenfunctions are to remain finite for x, what are the allowed eigenvalues?3. 3.27 Evaluate the commutators (a)x,px, (b) x,px2, (c) x,py, (d) x,V(x,y,z),(e) x,H, where the Hamiltonian operator is given by Eq. (3.45); (f) xyz,px24. 7.4 Let A and B be Hermitian operators and let c be a real constant. (a) Show that cA is Hermitian. (b) Show that A+B is Hermitian.5. 7.5(a) Show that d2/dx2 and Tx are Hermitian, where Tx-(2/2m)d2/dx2 show that Tx=22mx2d (c) For a one-particle system, does T equal Tx+Ty+Tz? (d) Show that T0 for a one-particle system.6. A25. 一维势箱（0，a）中的粒子的状态为 ，计算：能量的可能测量值及相应的几率；能量的平均值；求归一化系数A。7. 7.37 (a) Show that, for a particle in a one-dimensional box (Fig. 2.1) of length l, theability of observing a value of px between p and p+dp is 4N2s2ls2-b221-1ncosbldp (7.111), where snl-1 and bp-1. The constant N is to be chosen so that the integral from minus infinity to infinity of (7.111) is unity. (b) Evaluate (7.111) for p=nh/2l. At these values of px the denominator of (7.111) is zero and the probability reaches a large but finite value.Lecture 7: 角动量1. （5.17）(a) Show the three commutation relations (5.46) and (5.48) are equivalent to the single relation LL=iL. (b) find Lx2, Ly2. （5.24）Calculate the possible angles between L and the z axis for l=23. （5.25）Show that the spherical harmonics are eigenfunctions of the operator Lx2+Ly2.(The proof is short.) What are the eigenvalues?4. （5.30）Apply the lowering operator L- three times in succession to Y11(, ) and verify that we obtain functions that are proportional to Y10, Y1-1, and 0. 5.