Slide 1. Quantum Mechanics: the Practice

Transcription

1 Slde Quantum Mecancs: te Pactce

2 Slde Remnde: Electons As Waves Wavelengt momentum = Planck? λ p = = 6.6 x 0-34 J s Te wave s an exctaton a vbaton: We need to know te ampltude of te exctaton at evey pont and at evey nstant Ψ = Ψ, t

3 Slde 3 Wave Mecancs, t f t = Ψ t t t t t m Ψ = Ψ Ψ,,,, t t f f m = E m =

4 Slde 4 Statonay Scoednge s equaton E m a = m T H ˆ ˆ ˆ ˆ = = ˆ E H b = It s not poven t s postulated, and t s confmed expementally It s an egenvalue equaton Bounday condtons and egulaty must be specfed

5 Slde 5 Intepetaton of te Quantum Wavefuncton Ψ, t s te pobablty of fndng an electon n and t If =, t s sepaable: Ψ, t = f t = exp Et Remembe te fee patcle, and te pncple of ndetemnaton:f te momentum s pefectly known, te poston s pefectly unknown Ψ, t = Aexp[ k ωt]

6 Slde 6 Infnte Squae Well

7 Slde 7 Fnte Squae Well

8 Slde 8 A Cental Potental e.g. te Nucleus ˆ z y x H = = µ sn sn sn ˆ H = ϑ ϑ ϑ ϑ ϑ µ, ϑ lm Elm Elm Y = R E R R l l d d d d El El = µ µ

9 Slde 9 Solutons n a Coulomb Potental: te Radal Wavefunctons

10 Slde 0 Solutons n a Coulomb Potental: te Peodc Table ttp:// 5 d obtal

11 Slde <Bakets> > = = d δ = >= < * E d m H = >= < ˆ *

12 Slde aatonal Pncple E [ φ] = < φ Hˆ φ > < φ φ > E[ φ] E0

13 Slde 3 Electons and Nucle Hˆ = Tˆ ˆ ˆ ˆ ˆ H,..., = E e e e N N en,..., n tot n We teat only te electons as quantum patcles, n te feld of te fxed o slowly vayng nucle Ts s genecally called te adabatc o Bon- Oppeneme appoxmaton

14 Slde 4 Two-electon atom,, E Z Z =

15 Slde 5 Enegy of a collecton of atoms N-N : electostatc nucleus-nucleus epulson T e : quantum knetc enegy of te electons e-n : electostatc electon-nucleus attacton electons n te feld of all te nucle e-e : electon-electon nteactons ˆ e = ˆ en = I T RI ˆ ee = >

16 Slde 6 Mean-feld appoac Independent patcle model Hatee: eac electon moves n an effectve potental, epesentng te attacton of te nucle and te AERAGE EFFECT of te epulsve nteactons of te ote electons Ts aveage epulson s te electostatc epulson of te aveage cage densty of all ote electons

17 Slde 7 Hatee Equatons I I d R ε =,..., n n n L = Te Hatee equatons can be obtaned dectly fom te vaatonal pncple, once te seac s estcted to te many-body wavefunctons tat ae wtten as above as te poduct of sngle obtals.e. we ae wokng wt ndependent electons

18 Slde 8 Te self-consstent feld Te sngle-patcle Hatee opeato s selfconsstent! I.e., t depends n tself on te obtals tat ae te soluton of all ote Hatee equatons We ave n smultaneous ntego-dffeental equatons fo te n obtals Soluton s aceved teatvely

19 Slde 9 Iteatons to self-consstency Intal guess at te obtals Constucton of all te opeatos Soluton of te sngle-patcle pseudo- Scodnge equatons Wt ts new set of obtals, constuct te Hatee opeatos agan Iteate te pocedue untl t opefully conveges

20 Slde 0 Spn-Statstcs All elementay patcles ae ete femons alf-ntege spns o bosons ntege A set of dentcal ndstngusable femons as a wavefuncton tat s antsymmetc by excange,,...,,...,,..., =,,...,,...,,..., k n k n Fo bosons t s symmetc

21 Slde Slate detemnant An antsymmetc wavefuncton s constucted va a Slate detemnant of te ndvdual obtals nstead of ust a poduct, as n te Hatee appoac!,...,, n n n n n L M O M M L L ν β α ν β α ν β α =

22 Slde Paul pncple If two states ae dentcal, te detemnant vanses I.e. we can t ave two electons n te same quantum state

23 Slde 3 Hatee-Fock Equatons * * I I d d R λ µ µ λ µ λ µ µ µ λ ε = Slate n =,..., Te Hatee-Fock equatons ae, agan, obtaned fom te vaatonal pncple: we look fo te mnmum of te many-electon Scoednge equaton n te class of all wavefunctons tat ae wtten as a sngle Slate detemnant

24 Slde 4 Densty-functonal Teoy Conceptually vey dffeent fom Hatee-Fock vaatonal pncple on te cage densty In pactce, equatons ave te same fom, but fo te excange enegy obtaned fom te densty, not te wavefunctons It s exact n pncple, but appoxmate n pactce: dffeent foms fo te excange-coelaton densty: LDA, GGA, ybds Hatee-Fock excange densty-functonal coelatons