原子物理第四章课后习题.pdf

第 1 页 共 12 页 第四章第四章 原子的精细结构：电子的自旋原子的精细结构：电子的自旋 4.2.An atom is in a 2 3 2 Dstate.What is its magnetic momentand the possible values of thezcomponent of the magnetic moment z ? 试计算原子处于 2 3 2 D状态的磁矩及投影 z 的可能值。 Solution:For an atom in a 2 3 2 Dstate,we get: 1 212 2 ss 3 2 j 2l Expression of the g-Factor for a single electron: 22 2 13 2 3 113131314 22 35 22221225 22 j s sl lsl g j jj Magnetic moment: 3342 15 11 2255 jjBBB j jg The possible values of the magnetic moment z : , j zjjB m g For 3 2 j ,the values ofzcomponent of the total angular momentum: 31 , 22 j m 26 , 55 jj m g Hence, , 26 , 55 j zBB 4.4.In a Stern-Gerlach experiment,a thin beam of silver atoms which are in the 2 1 2 Sground state passes through a very inhomogeneous magnetic field and reaches a screen.The length of the magnetic field region is10.dcmThe distance between the center of the magnet and the screen is25.DcmThe speed and mass of each Ag atom are 400m/s and 107.87u,respectively.The interval distance between two neighboring lines in the spectrum after the splitting is 2.0mm.Find the value ofBz. 在施特恩-盖拉赫实验中，处于基态的窄的银原子束通过极不均匀的横向磁场， 并射到屏上，磁极的纵向范围10dcm，磁极中心到屏的距离25.Dcm如果银 第 2 页 共 12 页 原子的速率为400/m s，线束在屏上的分裂间距为2.0mm，试问磁场强度的梯度 值应为多大？银原子的基态为 2 1 2 S，质量为107.87 . u Solution: The distance 2 z: 2 2 z z BdD z z mv zcomponent of the magnetic moment z : zJJB m g For an atom in a 2 1 2 Sstate: 0L ;1 2S ;1 2J ;1 2 J m ;2 Js gg1 JJ m g Hence, 22 222 2 zzz zBB BBBdDdDdD zz z mvz mvz mv Substituting the following data: 441923 0.5788 10/0.5788 101.602 10/0.93 10/ B eV TJ TJ T 3 2 2.02 10zmmm 2 2525 10Dcmm 2 1010 10dcmm 400/vm s 326 23 107.87 107.87(107.87)1017.9 10 6.02 10 A muNgkgkg the value ofBz: 2 1.23 10/ z B T m z 4.6.In a Stern-Gerlach experiment,hydrogen atoms are emitted from an oven at a temperature of 400K.They are split into two beams by the time they reach the screen.They interval distance of these two lines is 第 3 页 共 12 页 0.60cm.If we substitute for hydrogen atoms chlorine atoms in their 2 3 2 P ground state,with the other experimental conditions remaining the same,into how many lines will the atomic beam of chlorine be split?What is the interval distance of the two neighboring lines? 在施特恩-盖拉赫实验中，原子态的氢从温度为400K的炉中射出，在屏上接受到 两条氢束线，间距为0.60cm。若把氢原子换成氯原子（基态为 2 3 2 P） ，其他实验 条件不变，那么，在屏上可以接受到几条氯束线？其相邻两束的间距为多少？ Solution:For chlorine atoms in their 2 3 2 Pground state: 1L ;1 2S ;3/ 2J ;3/ 2, 1 2 J m ; For a fixed value ofJ,the number of values of J mis214J , 4 lines. g-Factor: 22 2 1131314 222213 J S SL LSL g J JJ 2 2, 3 JJ m g 2 2, 3 zJJBBB m g 222 2 2, 3333 zzz zBB BBBdDdDdD zzz zkTzkTzkT For hydrogen atoms in their 2 1 2 Sstate: 0L ;1 2S ;1 2J ;1 2 J m ;2 Js gg1 JJ m g zJJBB m g 22 20.6 333 zzz zBB BBBdDdDdD zzcm zkTzkTzkT Hence, 22 0.6,0.2zcm zcm The interval distance of the two neighboring lines for chlorine atoms: 222 0.4zzzcm 4.7.The spectrum of a hydrogenlike ion exhibits the doublet structure of the principal lines in the Lyman series.Its difference in wavenumber is 29.6 1. cmWhich hydrogenlike ion is it? 试问波数差为 29.6 1 cm的莱曼系主线双重线，属于何种类氢离子？ Solution:A hydrogenlike ion exhibits the doublet structure of the principal lines in the Lyman series ,that is,the electron transits from n=2 第 4 页 共 12 页 ton=1,forn=1,theorbitalangularmomentum 2 1 2 01lS ,for n=2, 222 1 21 23 2 0,12,2,2lSPP .So the doublet structure of the principal lines comes from 21PS . According to the equation: 3 4 1 4 31 1 5.84 15.84 vn l lZ vcmZ n l lcm Substituting the following data: 1 29.6vcm , 2n ,1l Hence,Z=3,that is,the hydrogenlike ion isLi. 4.8.The electron in a hydrogen atom is in the 2P state.Estimate the magnetic field strength produced by the electron at the location of the proton.试估计作用在氢原子 2P 态电子上的磁场强度。 Solution:For the splitting of the 2P state into 22 1 23 2 2,2PP states in the hydrogen atom,we can calculate the energy difference between these two states is: 4 6 5 04 3 3 0.511 10 4.53 10 21 2 21 2137 Z UEeVeV n l l Or 4 45 3 7.25 104.53 10 1 Z UeVeV n l l According to the equation: 2 B UB The magnetic field strength is: 5 4 4.53 10 0.39 22 0.5788 10/ B UeV BT eV T 4.10.A zinc atomic spectral line ( 33 10 SP )splits because of the Zeeman effect in a magnetic field of 1.00BT .If we look at it in the direction perpendicular to the magnetic field,how many lines will be observed?What is the wavenumber difference between the two neighboring lines?Is it a normal Zeeman effect?Draw a diagram for the trasitions between the energy levels. 锌原子光谱中的一条谱线（ 33 10 SP ）在B为 1.00T 的磁场中发生塞曼分裂，试 第 5 页 共 12 页 问：从垂直于磁场方向观察，原谱线分裂为几条？相邻两谱线的波数差等于多 少？是否属于正常塞曼效应？并请画出相应的能级跃迁图。 Solution:For a zinc atom in a 3 1 S state: 1 22 1 2 2131,0,1,0, 1 31 2 22 SSLJm SL g J For a zinc atom in a 3 0 P state: 2 22 2 2 2131,1,0,0 313 222 SSLJm SL g J When there is a magnetic field B applied,the transition energy will be: 2121221121 2 0 2 2 0 2 BB B hvEEEEm gm gBEEB hvB This means that a spectral line will split into three lines. 222 111 000 224 222 B ee BvveBeB vvvv ccchcmcm c Wavenumber difference between the two neighboring lines is 19 1 318 221.602 101.00 93.4 2222 9.11 103.0 10/3.14 B ee Be BeBcT m hcmcm ckgm s , 0S , its not a normal Zeeman effect. 第 6 页 共 12 页 4.12.Fine structure will be produced when an electron in a potassium atom makes a transition from the first excited state to the ground state.The corresponding wavelengths are 7664A and 7699A .If we put the atom at a position where the magnetic field is B,the energy levels related to these spectral lines of fine structure will make a further split. (a) Give the interval distance of the splitting lines and draw a diagram of the energy levels after splitting. (b) The energy difference between the highest and the lowest is2 E .The initial difference is1 E .If21 1.5EE ,what is the value of B? 钾原子的价电子从第一激发态向基态跃迁时，产生两条精细结构谱线，其波长分 别为7664A 和7699A ，现将该原子置于磁场B中（设为弱场） ，使与此两精细结 构谱线有关的能级进一步分裂。 （a）试计算能级分裂大小，并绘出分裂后的能级图。 （b）如欲使分裂后的最高能级间的差距2 E 等于原能级差1 E 的 1.5 倍， 所加磁 场B应为多大？ Solution:(a) A potassium atomic spectral line ( 22 3 2,1 21 2 44PS )splits when there is no magnetic field. For a potassium atom in a 2 1 2 S state: 0 22 000 2 2121 2,0,1 2,1 2 31 2,1 22 ssljm sl gm g j For a potassium atom in a 2 1 2 P state: 第 7 页 共 12 页 1 22 111 2 2121 2,1,1 2,1 2 3121 , 2233 ssljm sl gm g j For a potassium atom in a 2 3 2 P state: 2 22 222 2 2121 2,1,3 2,1 2, 3 2 3142 , 2 2233 ssljm sl gm g j Interval distance of the splitting lines in 2 1 2 S , 2 1 2 P , 2 3 2 P are: 24 2, 33 BBB BBB (b) 21221111 617 333 BBB EEm gm gBEBEB 2121201021 12 2112 1.51.51.51.5 1.51.5 EEEEEEEEhvhv hchc hc 12 12 3 41 71 32 769.9766.431 1.24 1027.2 7 0.5788 102769.9 766.4 BB hc BeV nmnmT eV T 第 8 页 共 12 页 4.13.If the atom is in a position where the external magnetic fieldBis greaterthantheinternalmagneticfield,thetotalspinangular momentumSand the total orbit angular momentumLwill precess individually aroundB,respectively,andLandSwill no longer be coupled. (a) Write the expression for the total magnetic moment. (b) Write the expression forEin such an environment. (c) If the atom is sodium,calculate the energy splitting of the first excited and the ground state.Draw a diagram for the energy levels after the split,and point out which transitions are permitted according to the selection rules(0,0, 1 sl mm). 假如原子所处的外磁场B大于该原子的内磁场，那么，原子的L S耦合将解脱， 总轨道角动量L和总自旋角动量S将分别独立地绕B旋进。 （1）写出此时原子总磁矩的表达式； （2）写出原子在此磁场B中的取向能E的表达式； （3）如置于B磁场中的原子是钠，试计算其第一激发态和基态的能级分裂，绘 出分裂后的能级图，并标出选择定则（0,0, 1 sl mm）所允许的跃迁。 Solution:(a)Total magnetic moment : (2 ) 22 LS eee eee LSLS mmm (b)the expression forE: (2 )(2)2 222 (2)(2) 2 zzzls eee lslsB e eeBeB UBLSBBLSmm mmm e B mmmmB m (c) 2 1 2 3 S: 11 ,0, 22 slj, 1 0,21 2 lsls mmmm , 1B EB 2 1 2 3 P: 11 ,1, 22 slj 1 0, 1;,20, 1, 2 2 lsls mmmm , 2 (2,1,0, 1, 2) B Eu B According to the selection rules(0,0, 1 sl mm),we draw the diagram 第 9 页 共 12 页 as followed: 4.18.The transition(1 3s d) 1 2 D (1 2s p) 1 1 Pin a helium atom(singlet state)produces a wavelength6678A .Here1 3s dmeans one electron is in the1,0nlstate and the second electron in the3,2nlstate.They couple so their total spin is zero,then their totalLis that of the second 3d electron and the totalJL,so the state is 1 2 Dwhere now 21S J L is for totalS,LandJ.In a1.2BTmagnetic field,the Zeeman effect occurs.How many lines are there?What are their wavelengths and their polarization states for: (a)transverse observation?(b)longitudinal observation? Solution:(a)For state1 2: D 22 22 2 31 0,2,2,0, 1, 2,1,0, 1, 2 22 JJJJ SL SLJmgm gm g J For state 1 1 P : 22 11 2 31 0,1,1,0, 1,1,0, 1 22 JJJJ SL SLJmgm gm g J According to the selection rule 0,0, 1 sl mmm ,we draw the diagram as followed: 第 10 页 共 12 页 221121 1 0 1 BBB hvhvm gm gBhvmmBhvB 111 111 000 224 111 B ee BvveBeB vvvv ccchcmcm c There will be 3 lines.The wavenumber difference between the two neighboring lines is: 19 1 318 1.602 101.2 56.0034 44 3.14 9.11 103.0 10/ e eBcT m m ckgm s 1 2 2 10 3 3 10 1.0,6678 111 2 .1,6677.8 1 56.0034 6678 10 111 3.1,6678.2 1 56.0034 6678 10 B B mA B mmA hc B mmA hc Transverse observation:3 linearly polarized beams are seen.One is parallel to B(),the other two are perpendicular to B(, ) Longitudinal observation:2 circularly polarized beams are seen.One is right circularly polarization( ),the other one is left circularly polarization( ). 第 11 页