Download Notes on Time Dependent Perturbations in Quantum Mechanics | Phys 451 and more Study notes Quantum Mechanics in PDF only on Docsity! Chapter 13 Time Dependent Perturbations We take a look at time dependent potentials V(t) through a pertrbative approach. H = Ho +V (x,t)!!!!!!!!!!!!!!!!!!!!!!Time!dependent !Hamiltonian i! ! !t " (x,t) = H" (x,t)!!!!!!!!!!!!!Schrodinder !Eq.!!!!!!!! Let !!H" n = En" n !!!!!!!!!!!!!!!!!!!!!Unperturbed !eigenvalue!equation " (x,t) = cn n # (t)!" n (x)e$ i%nt !!!!!Generalized !solution i! { "cn n # (t)!" n (x)e$ i%nt $ i%ncn (t)!" n (x)e$ i%nt} = Ho +V (x,t)( ) cn n # (t)!" n (x)e$ i%nt !!!!! i! { "cn n # !" n !e$ i%nt $i En ! cn !" ne $ i%nt cancel # $%% &%% } = n # cnHo !" ne$ i%nt cancel # $%% &%% !+cn !V" n !e $ i%nt !!! i! { "cn n # !<" m |" n > &mn # $% &% !e $ i%nt} = n # cn !<" m |V |" n >!e$ i%nt ! !! "cmn (t)e $ i%mt != $i ! ! n # cn !<" m |V |" n >!e$ i%nt ! !! cmn (t)!=! $i ! n # dt 'cn !<" m |V |" n >!ei ! %m $%n( )t 0 t ' !!!!!!!!!!!!!!!!!!Transition!amplitude!between!state!m!(!n Perturbative!approach Let !!cn = cn (0) + cn (1) + .... cm (t)!=! !i ! n " dt '!(cn(0) + cn(1) + ..)!<# m |V |# n >!e! i$nt ' 0 t % 0th!order!! cn (0) = &ni !!!where!the!system!is!in!stationary!state!n = i 1st!order!!!with!cm (1) (0) = 0! cm (1) !=! !i ! n " dt '!cn(0) &ni " !!<# m |V |# n >!e i$mnt ' t0 t % !=!! !i ! dt ' t0 t % <# m |V |# i >!ei$mit ' cm (1) =!! !i ! dt ' t0 t % <# m |V |# i >!ei$mit ' !!!!!!!!!!Pmi (t) = cm(1) 2 c m (t) = ! mi + c m (1) (t) + ... perturbation series Harmonic!Perturbation! !V ( ! x,t) =!V ( ! x)!e ± i! t cm (1) !=!! !i ! dt ' 0 t " <# m |!V !|# i > e± i$ t !ei $mi ±$( )t = !i ! <# m |!V0 !|# i > dt ' 0 t " ei $mi ±$( )t ' Pm%i (1) != cm (1) 2 = 1 !2 <# m |!V !|# i > 2 dt ' 0 t " ei $mi ±$( )t ' 2 2&!t' (Em !Ei ±E( ) " #$$ %$$ Pm%i (1) != 2& ! !t ! <# m |!V !|# i > 2 ' (Em ! Ei ± E( )!!!!!!!! probability )m%i (1) = 2& ! ! <# m |!V !|# i > 2 ' (Em ! Ei ± E( )!!!!!!!!!! probability! per !unit !time Electromagnetic!Absorption!and!Stimulated!Emission pµ ! pµ + qAµ H = p ! eA( )2 2m + " +V = p 2 2m !! e m ! Aip radiation gauge! Ai p= pi ! A "#$ %$ + 4 e 2 2m ! Ai ! A neglect " #$ %$ + " !!!!!=0 radiation gauge & +V H = p 2 2m !+!V H0 "#$ %$ !!! e m ! Aip H '=V (t ) "#$ %$ Plane!Wave!Solution!!!!!!!! ! A = ! # ! ! A e i k ir±$ t( ) !!!!! ! # i ! k != 0!!!!!!!!! V (t) =!!! e m ! Aip = !! e m ! A ! # ip !eik ire± i$ t Pm%i (1) != 2& ' !t ! A 2 ! <' m |! e m ! # i ' i ! (!eik ir !|' i > 2 ) (Em ! Ei ± E* ) Pm%i (1) != 2&e2 ' !t ! A 2 ! ' i <' m |! ! # i ! (eik ir !|' i > 2 ) (Em ! Ei ± E* )!!!!!!!!!! ! A 2 = 2&c2N' $V !! + ,- . /0 Electric!dipole!approximation!!!!e ik ir ~!1+ ikir ! 1 2 kir( ) 2 + ...!! Pm"i (1) != 2#e2 ! !t " A 2 ! ! i <$ m |! " % i " &eik ir !|$ i > 2 ' (Em ! Ei ± E( ) Pm"i (1) != 2#e2 ! !t " A 2 !| ! i <$ m |! " % i " &(1+ ikir electric dipole # ! 1 2 kir( ) 2 forbidden transitions $%& '& + ...!) |$ i >| 2 ' (Em ! Ei ± E( ) k ε E B Stimulated Emission Absorption p p Aµ=(Φ ,A)
