《量子力学》第五章 近似方法（5.2）量子跃迁

5-2量子跃迁 ■当扰动依赖于时间时,就不是能级问题, 而是能级间的变化问题或跃迁问题

5-2 量子跃迁 ◼ 当扰动依赖于时间时，就不是能级问题， 而是能级间的变化问题或跃迁问题

含时微扰 H=Ho+H(t)=Ho+aw(t) )m)=Em m),(mn=8mn,mml iha p)=hy H()=∑,cn()e-"nx) ∑,cnem"Wm2() =(em-En)/h Wmn (t)=(mIN

含时微扰 W t m W n i c c e W t t c t e n i H H m m m n m m H H H t H W t m n m n m n m n i t t m n n i t n n t m m n m n n = = −  =  =   =  = = = = + = +    − ( ) ( )/ ( ) ( ) ( ) , , 1 '( ) ( ) / 0 0 0             

微扰展开 (t) (t) O dt (1) W. iomnt(o) (+1) ()

微扰展开 ( 1) ( ) (1) (0) (0) ( ) 0 ( ) ( )         n i t m n m n n i t m n m n m m m c W e c dt d i c W e c dt d i c dt d i c t c t m n m n    = = = = +   

初始条件和一级修正后波函数 t=0 (t=0)=k), Ek, cm k n)(t) mk (Temk dT Y()〉=|k())+∑Acm)(t)m() iat/ h 77t 77t

初始条件和一级修正后波函数 m t e m t k t c t m t W e d i c t t k c t i t m m i t m m k k m m k m m k   / (1) 0 (1) (0) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( 0) , , 0,         − =  = + =   = = = =  

T时刻处态mm≠k之几率 k 〈m|() 「"rnmk(z) LOLZ i

T时刻处态 之几率 2 0 2 (1) 2 ' ( ) 1 ( ) ( ) ( )  = = =  t i m k m m k H e d i c t P t m t m k       m ,m  k

跃迁速率 Wmk= transition prob. per unit time mk dt

跃迁速率 dt dP w transition prob per unit time m k m k = =

周期微扰 H(t)=AcoS(at+o) lot i(a+Omk )t i COmk -o)t Hmk= Fy mk ia+amk) mk (O mk ) max nun O=amk,em= Ek ho, absorption 力a. enZSSlOr

周期微扰 emission absorption imum i e F i e H F Fe F e H t A t m k m k m k m k m k i t m k m k i t m k m k i t i t m k m k , , , , max ( ) 1 * ( ) 1 ' ˆ ˆ cos( ) ˆ '( ) ( ) ( )                         = − = − = = + − − + + − = = + = + + − + −

吸收跃迁 k ei(a-amk ) Pmk (t) z k) k zt sin 2[Co-omk t/21 a[(c-cm,k)/2卫2 ÷年/ Em一(Ek十c

吸收跃迁 ( ) [ ( )] 2 [( )/ 2] sin [( ) / 2] ( ) 1 1 ( ) 2 2 2 2 2 2 ( ) →   − + − − = − − =  − t F t t F t t i e F i P t m k m k m k m k m k m k i t m k m k m k m k                     

费米黄金规则 dP2丌|F k mk mk SEm-( tha For emission E mk 2m上m8 (Ek -)

费米黄金规则 [ ( )] 2 [ ( )] 2 2 2               = − − = = − + m k E m k m k m k A m k m k m k F w For emission F dt dP w

电偶极跃迁 h=ey=eE. x=-D. Eo cost D=ex(e=1/3

电偶极跃迁 cos 1/ 3 ( , ) cos [ ( )] 2 ( ) [ ( )] 2 ( 0) ' cos 2 0 2 2 2 0 2 2 0 0  = = = − +   − + =  = = −  = −                 angle of D E x e E w D E Absorption D ex e H e eE x D E t m k m m m k m k m m               