量子力学分波法

ifkfkTkVkfi|ˆ|π4||π4),()(22)(22lllififiTlkkkT)(cosP4π12)(12llliSlk)(cos)P11)(2(π8i22llliSlkf)(cos)P11)(2(2i),(lllllk)(cosPesin1)2(1i)(llfllllllllkll)(cosP)(cos)Pcos(sinsin1)1)(2(2),(2lllk22sin)12(π4lltlk22sin)12(π4ll光学定理)0(Imπ4llfk)0(Imπ4fkt分波法ararVrV0)(0例：球方势阱和球方势垒的低能散射。入射粒子能量很小，其德布罗意波长比势场作用范围大很多。0)()1()(d)(ddd12222rRrllrUkrrRrrrll222)(2)(,2rVrUEkararkrVrU0)(2)(202V0是势阱或势垒的深度或高度解:1kaa只讨论s分波arrRkrrRrrr0)(d)(ddd102022arrRkrrRrrr0)(d)(ddd1020222022kkk2022kkkrrurR)()(000)(d)(d02202rukrru0)(d)(d02202rukrruarkrBruarrkAru)sin()()sin()(0000arkrrBrRarrkrArR)sin()()sin()(00000)0(0u0rrrurR)()(00有限00ar)(0ru)(0ru和连续akkkaktg1)(tg10kaakkk)tg(arctg00220sinπ4kllt])tg(arctg[sinπ422kaakkkkarrRkrrRrrr0)(d)(ddd102022arrRkrrRrrr0)(d)(ddd102022若粒子能量很低0kk1)1tg(tg)tg(arctg000akakkakaakkkkaakkk1,arctgxxx022sinπ4kt2002202)1tg(π4π4akakak注：00U若，前面结果中00ikk)1th(000akakka2002)1th(π4akakat0V0k1th00000akakakakeeeeak2π4at低能粒子被无限高势垒散射，总散射截面为经典的4倍kaakkk)tg(arctg0§5含时散射理论一、含时格林算符)(|ˆ)(|itHttiippip|是动量入射时的散射态ip)(|)()(|ˆi0trVtHtiipp)()(ˆˆi00ttttGHt引入格林算符定态散射过程区域很小0)(rVtt0t用含时薛定谔方程处理，更具一般性。)(rVV)(ˆi)(000e)(θi)(ˆˆttHttttGG)(ˆi)(000e)(θi)(ˆˆttHttttGG两组解0001)(θttt)(d)(dθttt)()(ˆˆittttGHt类似地，可以引入全格林算符)()(ˆˆi00ttttGHt)(ˆi)(e)(θi)(ˆttHttttGttrVttGttiiipppd)(|)()(ˆ)(|)(|)()(0)(ttrVttGttiiipppd)(|)()(ˆ)(|)(|)()(含时薛定谔方程的解：或写成：其中：0)(|)ˆ(i0tHtiptEipiiktie|)(|)(ˆi)(00e)(θi)(ˆttHttttGdiπi21lim)(i0xexdeieπ21)(ˆ)(ˆi)(i)(00ttHttttG两边乘以对t积分得：)(iettEddeeieπ21de)(ˆ)(i)(ˆi)(i)(i)(00ttttGttEttHttttEdi)ˆ(0/HE/iˆ10HEiˆ10HE无穷小整数)(ˆi)(e)(θi)(ˆttHttttG)(ˆi)(00e)(θi)(ˆttHttttGddei1π21)()ˆ(i0tttHEiˆ1de)(ˆ0)(i)(0HEtttGttE)(ˆ)(0EG互为傅里叶变换！！EHEttGttEdeiˆ1π21)(ˆ)(i0)(0EEGttEd)e(ˆπ21)(i)(0同理EHEttGttEdeiˆ1π21)(ˆ)(i)(EEGttEd)e(ˆπ21)(i)(ttrVttGttiiipppd)(|)()(ˆ)(|)(|)()(ttrVEEGtiipttEpd)(|)(d)e(ˆπ21)(|)(i)(tEipiiktie|)(|EtkrVEGktitEttEtEipiiidd|)()(ˆeeπ21e|)(|)(i)(ii)(EtkrVEGktitEttEtEipiiidd|)()(ˆeeπ21e|)(|)(i)(ii)(时0tEkrVEGtkitEEipiid|)()(ˆdeπ21|)0(|)()i)(（EkrVEGEEkiiid|)()(ˆ)(|)(iiikrVEGk|)()(ˆ|)()(|ip正是前面定态散射处理下得到的LS方程)(ˆi)(e)(θi)(ˆttHttttG时t)(ˆi)(e)(θi)(ˆtHttG0时t)(ˆi)(e)(θi)(ˆtHttG00)(|)(|lim)(ttiippt0)(|)(|lim)(ttiippt定态解0)(|)(|lim)(ttiippt时t)(|)(|)(ttiippt增加，第二项的影响逐渐出现。tt的态)(|)(tip)(|)(tip通过V影响当前的态未来的态()不影响当前的态tt格林算符：传播子、传波函数)(ˆi)(e)(θi)(ˆttHttttG推迟格林算符类似的讨论可知，表示的是它传递到当前的信息是来自于未来时刻()的，它称为超前格林算符，显然并没有清晰的物理含义。)(ˆi)(e)(θi)(ˆttHttttGttttrVttGttiiipppd)(|)()(ˆ)(|)(|)()(这种影响由传递)(ˆ)(ttG)(ˆi)(e)(θi)(ˆttHttttG它也是薛定谔方程的解，其演化规律是确定的，也是可以存在的。（与不同）)(ˆ)(ttG实际上：†)(ˆ)(ˆˆTttGT†)(ˆiˆe)(θiˆTttTttH)(ˆie)(θittHtt)(ˆ)(ttG互为时间反演算符)(|ˆ)(tTipttrVttGttiiipppd)(|)()(ˆ)(|)(|)()(ttrVttGttiiipppd)(|)()(ˆ)(|)(|)()(ttrVttGTtTiippd)(|)()(ˆˆ)(|ˆ)(ttTrVttGtiippd)(|ˆ)()(ˆ)(|)()(|)(tip)(|)(tip表示的是的时间反演态)(|)(tip二、摩勒算符（）和散射算符（）ΩˆSˆ)(|ˆ)(|itHttiippttVttGttiiipppd)(|)(ˆ)(|)(|)()(0)(含时散射态)(|),(ˆ)(|)()(tttUtiippHttttUˆ)(ie),(ˆ时间演化算符ttVttGttpppd)(|)(ˆ)(|)(|)()(0in)(ttVttGttpppd)(|)(ˆ)(|)(|)()(0out)(0)(|)ˆ(iin0tHtpEtpktiine|)(|Etpktioute|)(|0)(|)ˆ(iout0tHtp)(|),(ˆ)(|in0intttUtpp)(|),(ˆ)(|out0outtttUtpp0ˆ)(i0e),(ˆHttttU)(|),(ˆ)(|in0intttUtpp)(|),(ˆ)(|out0outtttUtpp时t0)(ˆ)(0tG时t0)(ˆ)(0tG)(|)(|in)(ttptp)(|)(|out)(ttptp)(|intp为的射入渐进态)(|)(tp)(|outtp为的射出渐进态)(|)(tp含时自由态)0(|ˆ)0(|in)()(ppΩ定义：)0(|ˆ)0(|out)()(ppΩ)0(|)0,(ˆ)0(|)0,(ˆin0)(ptptUtU)0(|)0,(ˆ)0,(ˆ)0(|)0,(ˆ)0,(ˆin01)(1ptptUtUtUtU)0(|)0,(ˆ),0(ˆ)0(|in0)(ptptUtU)0,(ˆ),0(ˆlimˆ0)(tUtUΩt)0,(ˆ),0(ˆlimˆ0)(tUtUΩt同理)0,(ˆ),0(ˆlimˆ0)(tUtUΩt)0,(ˆ),0(ˆˆ0)(UUΩ)0,(ˆ),0(ˆˆ0)(UUΩ0ˆiˆi)(eelimˆHtHttΩ)0(|)0(|ˆ)(in)(ppΩ)()(||ˆkkΩ)()(|)0(|ppEtpktiine|)(|定态解kp|)0(|inEtpktioute|)(|与定态求解时的定义一致)(ˆΩ是不含时的算符散射算符：)(†)(ˆˆˆΩΩS)(i