Cem 564 Lecure 8 3mar From Noes 8 003,005,007, 009 TIME IN QUANTUM MECANICS. I Ouline I. Te ime dependen Scroedinger equaion; ime dependence of energy eigensaes II.. Sae vecor (wave funcion) ime evoluion A. 3 cases of a -sae problem B. cases of linear combinaion of infinie number of eigensaes: wave paces ) an oscillaing wave pace (model for a laser); gaussian pace in parabolic well ) a free "paricle": propagaion of a gaussian wave pace under no force. III. Direc evoluion of e densiy: Te Liouville Operaor I. Energy Eigensaes ave Saionary Densiy (are saionary saes) Te ime dependen Scroedinger equaion, Ψ( x, ) ˆ Ψ( x, ) is arguably e mos fundamenal equaion discovered by umanind. I predics e fuure providing we now e presen. Aloug we canno now e wave funcion of e world, be can in cerain experimens now e wave funcion of a small isolaed par of e world and use is equaion o undersand e resuls of e experimen. If Ψ EΨ, en e above equaion says Ψ( x, ) E Ψ( x, ) and e soluion is: Ψ or ( x, ) Ψ( x,0) e E Ψ Ψ E ( Ψ ( x, ) Ψ ( x, ) e. Tus, a any ime,, Ψ Ψ( x, ) ( x, ) f( x) i.e., is independen of ime. )
Cem 564 Lecure 8 3mar II. Time Dependence of Wave funcions (wo-level sysem) Recipe: ) Use Scroedinger Eq. o ge Ψ. (inegrae dψ () ) ) ge densiy from Ψ*() Ψ (). Le Θ > c () > + c () >, were > and > depend on space only. Ten, Ψ c() c() ) ) + c() i c() Ge wo equaions by muliplying on e lef by < and by <. c i c i () () c () c i c i () () c () Scroedinger equaion. Tis is e exac wo-sae form of e We nex examine ree special cases. Te general resul is a mixure of ese ree cases. Case : Resonance, real ineracion: > 0 ; 0 c i () c () c i () c () Te soluion is : c () cos( ) cos( ω ) c () isin( ) sin( ω ) wi ω as can readily be demonsraed by subsiuion ino e wo coupled equaions above.
Cem 564 Lecure 8 3mar 3 Evoluion if c (0) and c (0) 0 c : 0-0 c : 0 -i 0 i 0 We see a e resul is a periodic reurn o e iniial sae--e caracerisic of resonance. In erms of e sae funcion (vecor): i i + i i i + i Tis resul says a e sronger e ineracion e iger e frequency and e sorer e period of oscillaion. Tis is e fases possible rae for moving from sae > o sae >. Te ime required o cange compleely from sae > o sae > is e ime suc a /ћ π/, i.e., wen π/ 4 A pysical example of suc a resonance is an excied aom nex o an unexcied one. A* B A B* As depiced, e ineracion can be roug space (coulombic), called resonance energy ransfer, or e exciaion could be ransfered by double elecron excange creaed by orbial overlap, or BOT. A* B A B*
Cem 564 Lecure 8 3mar 4 Densiy Marix Evoluion In e end, wa couns is e ow e probabiliy densiy canges in ime. Densiy is given by Ψ Ψ ( c + c )( c + c ) cc + cc + cc + cc, wic defines e densiy marix, cc cc ρ() cc cc cos ( ω) icos( ω) sin( ω) icos( ω) sin( ω) sin ( ω) wic evolves in e periodic manner: 0 0 0 05. i05. 0 0 05 05 i.. 0 05. 05. 0 05 05 i.. 0 0 Case : Resonance, imaginary ineracion: - ib, b>0 ; 0 Same as case excep for e pase: c ( ) cos(cos( ω ) c () sin( ω ) wi ω Te evoluion of e densiy marix in is case is: 0 0 0 05. 05. 0 0 05 05 0 05 05 0.. 05 05.... 0 0
Cem 564 Lecure 8 3mar 5 Case 3: Non-resonance, no ineracion : 0 ; 0 Tis is a simple linear combinaion of wo energy eigensaes. Tis resuls in wo uncoupled equaions. c i c i E () () c() c i c i E () () c () Te soluion of eac is jus as in e opening lines of is andou. owever, now we will loo o see ow e densiy canges in ime. Noe a e diagonal is ime independen, i.e., e probabiliies o be in e saes > and > are consan in ime if 0. owever e densiy does oscillae. Consider a 50% mixure a ime 0: Ψ ( 0) c() 0 c() 0 c() 0 c() 0 e ρ ( E E) * * c () 0 c () 0 e c () 0 c () 0 c () c () 0 e,, + ( E ) ( E E) * * ρ c ( 0) c ( 0) + c ( 0) c ( 0) + c ( 0) c ( 0) e + c ( 0) c ( 0) e If < s and < p and bo are real, ρ 0.5(s + p ) +cos(ω ) (ss), were ω (E p - E s )/ħ. Wen ω 0, e funcion is a ybrid orbial poining o e rig: and wen ω π, e funcion is a ybrid orbial poining o e lef: ( E E ) ( E E) ( E E) + i ρ( ) + e + e ( E E )
Cem 564 Lecure 8 3mar 6 In is case ere is an oscillaing dipole. Tis oscillaing dipole (called e elecric dipole ransiion momen or ransiion dipole) is wa ineracs wi lig during absorpion and emission. Is magniude deermines e rae of absorpion of lig. Wenever wo saes differ by one node as in is case, e ransiion dipole will be large and e ransiion will be srongly allowed. For a general superposiion of energy eigensaes: ( E ) Ψ () c ( 0) e m, ( E Em) * c () 0 c () 0 e m m ( E) m ( Em) i ω m * ω m c( 0) c( 0) + ( c( 0) cm( 0) e m + cm() 0 c() 0 e m ), * ρ Ψ() Ψ() ( c () 0 e )( c () 0 e m) were w m ( E Em) < m Noe a e wo erms in e second summaion are complex conjugaes, so e densiy is real, as i mus be. Eac erm in e second sum modulaes e saic average densiy given by e firs sum. Te Fourier ransform of e densiy would reveal one erm for every disinc energy difference beween e various energy eigenvalues represened. m B. Wave Paces. Coeren superposiion of armonic oscillaor wave funcions: Te oscillaing Gaussian wave pace. For e armonic oscillaor eigenvalues: E n (n +½) ħω. Consider e idealizaion of a very sor and inense laser pulse a elecronically excied a diaomic molecule wose excied sae vibraional poenial minimum is displaced from e ground minimum by an amoun d. Ten λ ½d / ħω
Cem 564 Lecure 8 3mar 7 Te vibraional funcion creaed in erms of e armonic oscillaor eigensaes on e upper surface are given by Ψ () e n/ / λ λ n 0 n! e i( n+ ) ω Noice a e squares of e coefficiens are e now familiar Franc-Condon facors (Poisson disribuion formula). Te densiy arising from is wave funcion as an amazing beavior. Te modulaing cross erms conspire o eep e gaussian sape of e iniial probabiliy densiy a all imes, bu is cener moves in ime precisely as would a classical paricle of e same mass in a poenial saring wi zero velociy a a poin! n
Cem 564 Lecure 8 3mar 8. Gaussian Superposiion of Momenum Eigenfuncions: Moion of A Free "Paricle" Below is sown an x-momenum eigenfuncion being operaed upon by e x-momenum operaor: i x e ix e ix Te energy of ese saes is E p /m ħ /m. Terefore e wave funcion wi is ime facor is given by: i m i( x ω ) Ψ ( x, ) e ix e e A superposiion of momenum eigenfuncions is erefore: Ψ () c e i( x ω ) If e c form a gaussian funcion, c ( ) a ( ) e 4 0 wi infiniesimally spaced values, e above summaion becomes e inegral below: 4 i( x ω ) Ψ ( x, ) N e e d a a ( ) 0 Tis inegral is sraigforward, and leads o e following expression for e probabiliy densiy (see Coen-Tannoudji, pp 6-65): ΨΨ * N a () e 0 a ( x ) m 4 4 a + m i.e., e densiy is a gaussian wic moves in e x direcion wi velociy ħ 0 /m and wose wid is increasing in ime. Te normalizaion consan erefore also increases in ime. Wiou e parabolic poenial energy well o eep refocusing e pace, i spreads wi increasing ime.
Cem 564 Lecure 8 3mar 9 III. Direc ime dependence of e DENSITY: Te Liouville Operaor Tae e ime derivaive of e densiy operaor: ( ) ρ $ () () () () () () Ψ Ψ Ψ Ψ + Ψ Ψ, ( ) i $ bu Ψ Ψ() adj $ and, Ψ() i Ψ() so, ρ i $ $ $ Ψ() Ψ() Ψ() Ψ() + i Ψ() Ψ() ( ) ρ i ρ ρ i $ $ $ $ [ $ i, ρ $ ] L $ ρ $ adj were L e Liouville Operaor or e Liouvillian and e above equaion is nown as e Liouville Equaion. I is equivalen o e Scroedinger Equaion in is form, bu if one allows e bras and es o evolve separaely in ime, i is more powerful. ρ i $ $(,') Ψ() Ψ(') Ψ() Ψ(') + i Ψ() Ψ(') $ adj Te laer form is necessary o properly describe cerain penomena.