《高等量子力学》课程教学课件（PPT讲稿）第六章 散射理论 6.4 分波方法

2V一阶波恩近似条件F平面波：2m对具确定角动量的态受散射的影响感兴趣分波法

§6.4 分波方法 一、自由粒子态：平面波与球面波 ◼ 两种选择： ◼ 平面波： ；球面波： |E, l, m> ◼ 对具确定角动量的态受散射的影响感兴趣→分波法 2 [ , ] 0, [ , ] [ , ] 0 z H p H L H L = = = 2 2 , 2 k E k k m = = 2 2 2 2 2 2 2 ( ) 2 2 2 p L H m m r r r mr   = = − + +   2 2 4 ( ', ) ' m f k k k T k  = − 大 ( ) 3/2 1 [ ( ', ) ]; (2 ) r ikr ikz e x e f k k r   + → + 2 0 2 0 2 | | 一阶波恩近似条件 1(低能）或 2 | | ln( ) 1(高能）并不总能满足 m V a m V a ka k  

、球面波的动量与坐标空间波函数：动量空间波函数(KE,1,m)=[(E,1,mK)][(E, 1, m|D(g, 0, 0)|k2))*=[Z[ dE'(Elm|D(0,0, 0)[E'1'm")(E'1'm'|k2)]D(0)*(p, 0, 0) [(Elm = 0| k2)]*4元Y"(h) [(Elm = O| k2)* = gie(k)Y"(k21+(K- E)(K[E, 1, m)(K[(H - E)|E, 1, m) = 0 :2mh~k2Eg1e(k)2m

二、球面波的动量与坐标空间波函数： ◼ 动量空间波函数 * * * ' ' ( )* * 0 * , , [ , , ] [ , , ( , ,0) ] ˆ [ ' ( , ,0) ' ' ' ' ' ' ] ˆ ( , ,0)[ 0 ] ˆ 4 ˆ ˆ ( )[ 0 ] ( ) ( ) ˆ 2 1 l m l m m m l lE l k E l m E l m k E l m D kz dE Elm D E l m E l m kz D Elm kz Y k Elm kz g k Y k l       = = = = = = = = +   2 2 2 2 ( ) , , 0 ( ) , , 2 ( ) ( ) 2 lE kl k k H E E l m E k E l m m k g k N E m  − = = − = −

据：S(E-E)S,.S'1'm'[Elm)=「dk(E'1'm'|k)(k|ElmHhk2h2k2= J dk [Nk s("- E)Y"(K)Y"**(K)2m2mmk= N s(E - E')Sh?h?k?h得：gie(k)H22 mVmkh8(E,- E)Y"(k)(k|Elm)=VmkhK)=Z[dE|Elm)(Elm|k)=ZZElmym*2h?k2mk1=0 m=-1FIm2m球面波对研究粒子衰变过程也特别有用（角动量守恒）

得： 据： ◼ 球面波对研究粒子衰变过程也特别有用（角动量守恒） ' ' 2 2 2 2 2 '* ' 2 2 ' ' ( ') ' ' ' ' ' ' ( ') ( ) ( ) ( ) ˆ ˆ 2 2 ( ') ll mm m m kl l l kl ll mm E E E l m Elm dk E l m k k Elm k k dk N E E Y k Y k m m mk N E E         − = = = − − = −   2 2 ( ) ( ) 2 lE k g k E mk m = −  2 2 * 0 2 ˆ ( ) l m l k lm l m l E m k dE Elm Elm k Elm Y k mk  = = − = = =     ˆ ( ) ( ) m k l k Elm E E Y k mk = − 

坐标空间波函数(x|Elm)=「dk(x|)《|Elm)hk- E)Y"(K)[2dkd(x|)2mmkY"(h) 2/4 m3/4E1/4exp(ik.x)oik.x= Z (21 + 1)i'j;(kr)P(k . x)[d?,福一h3/2(2元)3/21Z(21'+ 1)i'j,(kr) d2,P,(K.)(R)mk(2元)3/2h/Vmk4i'j,.(kr)[ d,**()y"(x)"(k)(2元)3/2 hmil2mk(k = 2mE /n)j,(kr)y"(x)h元h(k|Elm)S(E)-E)Y"(k)Vmk

◼ 坐标空间波函数 2 2 2 1/4 3/4 1/4 3/2 3/2 3/2 ' ' ' '* ' ' ' ' ' ' ˆ ( ) ( ) 2 exp( ) 2 ˆ ( ) (2 ) 1 ˆ ˆ (2 ' 1) ( ) ( ) ( ) ˆ (2 ) ˆ ˆ 4 ( ) ( ) ( ) ( ˆ m k l m k l l m l l l k l l m m m l l l l k l m x Elm dk x k k Elm k k dkd x k E Y k mk m ik x m E d Y k mk l i j kr d P k x Y k i j kr d Y k Y x Y     = =  −  =  = +   =         ( / ) 3/2 ) (2 ) 2 ( ) ( ) 2 ˆ l m l l mk k i mk j kr Y x k mE   = = ( ) ( ) m ˆ k l k Elm E E Y k mk = −  ˆ (2 1) ( ) ( )ˆ ik x l l l l e l i j kr P k x  = +  

h三、分波展开（对球心势）- E)Y"(kS(E.4lnVmk1T-V+V.1E-Ho+ie4元2m(K'|T|K)fCkh?4元2mZdEdE'《'[E'1'm")《E'I'm'|T|Elm)《Elm|)h?lm,1m24元ZT(E)Y"(K')Y"*(k) = f(0)k1m-T(E), =Z(k,')Z (21 + 1)f,(k)P(cos 0)(f(k) = -k

三、分波展开（对球心势） ( = 2 2 2 2 , ' ' 2 * 4 ( ', ) ' 4 ' ' ' ' ' ' ' ' 4 ˆ ˆ ( ) ( ') ( ) ( ) ˆ ˆ (2 1) ( ) (cos ) ( ) ( )， ( , ') lm l m m m l l l k lm l l l l l m f k k k T k m dEdE k E l m E l m T Elm Elm k T E Y k Y k f k l f k P f k T E k k k        = − = − = − = = + = −      ˆ ( ) ( ) m k l k Elm E E Y k mk = − 

三、分波展开（续）(f(k) = - " T,(E)f,(0) = Z(21 + 1)f(k)P(cos 0)keik.xE(21 + 1)i'j,(kr)P(Xi(kr_1元)-i(kr-1元22ee大rj,(kr)2ikr大rk=kze-i(kr-l元)LK1[Z (21 + 1)P(cos 0)(2元)3/22ikr1ikZ (21 + 1)f(k)P(cos 0)-i(kr-1元)ikr1Z(21 + 1) P(cos 0)e?{[1 + 2ikf,(k)](2元)3/22ikrrI散射体改变了出射球面波：1 -> 1+2ikf(k)

三、分波展开（续） ( ) (2 1) ( ) (cos ) ( ) ( )) ( k l l l l l f l f k P f k T E k    = + = −  ( ) 大 ˆ 3/2 ( ) 3/2 ( ) 3/2 1 [ ( ) ] (2 ) 1 [ (2 1) (cos )( ) (2 ) 2 (2 1) ( ) (cos ) ] 1 (cos ) (2 1) {[1 2 ( )] } (2 ) 2 ikr r ikz k kz ikr i kr l l l ikr l l l ikr i kr l l l l e x e f r e e l P ikr e l f k P r P e e l ikf k ik r r           + = − − − − ⎯⎯⎯⎯⎯→ + − = + + + = + + −    散射体改变了出射球面波： 1 1 2 ( ) -> l + ikf k ( ) ( ) 2 2 大 (2 1) ( ) ( ) ˆ ˆ ( ) 2 ik x l l l l l l i kr i kr r l e l i j kr P k x e e j kr ikr    − − − = +  − ⎯⎯⎯⎯→ 

f,(0) = Z (21 + 1)f(k)P(cos 0)四、分波相移e-i(kr-1)ik1Z(21 + 1) P(cos 0)O([1 + 2ikf,(k)](2元)3/22ikrL1由定态理论之概率流知向外与向内的通量相同。另据角动量守恒知：S,(k) = [1 + 2ikf(k) = 1（1分波的幺正性关系）e2is,eio sin S,-1S, -1（约定相移为28）f,2ik2ikkiIm(k01元exp(-i + i28,)kf+1222Z(21 + 1)e sin 8,P(cos 0)f (0)k1oRekfo图6.8的盗根图，OP是起，的大小，CO和CP均为单位四上长度为1/2的半径：角OCP=28折

四、分波相移 2 2 ( ) 1 2 ( ) 1 l l S k ikf k = + = ( 2 2 1 sin 1 约定相移为 ） 2 2 1 exp( 2 ) 2 2 2 l l i i l l l l l l S e e f ik ik k i kf i i       − − = = = = + − + 1 ( ) (2 1) sin (cos ) l i k l l l f l e P k     = +  由定态理论之概率流知向外与向内的 通量相同。另据角动量守恒知： ( ) ( ) 3/2 1 (cos ) (2 1) {[1 2 ( )] } (2 ) 2 ikr i kr l l l l P e e x l ikf k ik r r     − − + = + + −  ( ) (2 1) ( ) (cos ) k l l l f l f k P   = +  （l 分波的幺正性关系）

≥(21 + 1)e sin 8,P(cos 0)f(e) =五、散射截面kαtot = [f(0)'dQ[ d2 (21 + 1) (21+ 1) 2e-or sin, sin,P(cos0),(cos)2114元Z(21 + 1) sin° 8, = Zojeig sink21ks,～0，f,～纯实,儿,～，～纯虚，，～4元(21+1）（共振）kOtotZ(21 +1) sin28,Im f(θ = O) =(光学定理)4元K

五、散射截面 1 ( ) (2 1) sin (cos ) l i k l l l f l e P k     = +  ' 2 2 ' ' ' 2 2 ( ) (2 1)(2 ' 1) sin sin (cos ) (cos ) 4 (2 1)sin l l tot i i l l l l ll l l l l f d l l d e e P P k l k            − =  + + =  = + =      ( ) 1 2 Im ( 0) (2 1)sin 光学定理 4 tot l l k f l k     = = + =  2 0， 纯实， ， 纯虚， 4 (2 1) （共振） 2 l l l l l f f l       + sin l i l l e f k   =

六、相移的确定i(kr-1元1P,(cos 0)K大r128(xy(+)Z(21 + 1)Ak=kz(2元)3/22ikrr1-1元）-i(kr_1元)i(kr128,121eeNi'(21P(cos 0)(2元)3/222ikrikr1128,12Z (21 + 1)(h(2)(kr))P(cos 0)kr(2元)3/22211Z i(21 + 1)A,(kr)P,(cos 0)(2元)3/21A(kr) = em, (kr) in(k)+ (kr)_in(k) = e[eos ;j;(kr) -sino,n,(kr)]22d In(A,)ji(kR) cos S, - n,(kR) sin 8,dAPβ, d ln(r)dij,(kR) cos 8, - n,(kR) sin 8d"u2m1(1 + 1),k2求解球心势的径Ju, = O; u, = rA,(r); u, Ir=o= 0h2dr2向方程：kRj,(kR) - β,j,(kR)tans,并利用β,在r=R处的连续性，求得相移kRn,(kR) - β,n,(kR)

六、相移的确定 2 ( ) ( ) ( ) ( ) ( ) [cos ( ) sin ( )] 2 2 l l i i l l l l l l l l l j kr in kr j kr in kr A kr e e j kr n kr     + − = + = − ( ) ( ) 大 2 ˆ 3/2 ( ) ( ) 2 2 2 3/2 2 (1) (2) 3/2 3 / 1 (cos ) (2 1) { } (2 ) 2 1 1 (2 1){ } (cos ) (2 ) 2 2 1 1 ~ (2 1){ ( ) ( )} (cos ) (2 ) 2 2 1 (2 ) l l l ikr i kr l r l i k kz l l l i kr i kr i l l l i l l l l l P e e x l e ik r r e e e i l P ikr ikr e i l h kr h kr P               − − + = − − − ⎯⎯⎯⎯⎯→ + − = + − + + =    2 (2 1) ( ) (cos ) l l l l i l A kr P +  ' ' ln( ) ( )cos ( )sin | ( ) ln( ) ( )cos ( )sin l l l l l l l r R r R l l l l l d A dA j kR n kR r kR d r A dr j kR n kR      = = − = = = − 求解球心势的径 向方程： u u 2 2 2 2 2 0 2 ( 1) [ ] 0; ( ); | 0 l l l l l r d u m l l k V u rA r dr r = + + − − = = = 并利用l在r R = 处的连续性，求得相移 ' '( ) ( ) tan ( ) ( ) l l l l l l l kRj kR j kR kRn kR n kR    − = −

rR0,tan, =j(kR)A,(kr) = ei[cos 8,j,(kr)-sin 8,n,(kr)] = O;n,(kR)sin kR / kRtan = = -kR- tan kR,1= cos(kR) / kR1coskrA-o(kr) ~ sin krsin.cos S.sin(kr0krkrkrTAro(r)对低能散射（kR<<1)：(kr)(21 - 1)!!j,(kr)7音(21 + 1) !!'(kr)/+1-(kR)21+10tan,R=-dolk(21 + 1) [(21 - 1) ! ! J2图6.9rAr（r）对的图（因子e被去掉了）V=0时的虚线行为如同sinkr.实线是S波硬球散射的结果，与V=0时的情况比较，它被平移了R=一%/k.sinSdo=l f(0) P=R2=4元R2a.tot2dp

七、硬球散射 ( ) ( ) [cos ( ) sin ( )] 0; tan ( ) l i l l l l l l l l j kR A kr e j kr n kr n kR  = − = =    对低能散射（kRR , 0, V  = 0 0 0 0 0 0 sin / tan tan , cos( ) / sin cos 1 ( ) ~ cos sin sin( ) l kR kR kR kR kR kR kr kr A kr kr kr kr kr      = = = − = − − + = + 1 ( ) (2 1)!! ( ) ,n (kr) - (2 1)!! ( ) l l l l kr l j kr l kr + − + 2 1 2 ( ) tan ~0 (2 1)[(2 1)!!] l l kR l l  + − + − 2 2 2 2 0 2 sin | ( ) | ; 4 tot d f R R d k   = = =    