Identical Particles. We would like to move from the quantum theory of hydrogen to that for the rest of the periodic table

Transcription

1 We wuld like t ve fr the quatu thery f hydrge t that fr the rest f the peridic table Oe electr at t ultielectr ats This is cplicated by the iteracti f the electrs with each ther ad by the fact that the electrs are idetical The Schrdiger equati fr the tw electr at ca ly be slved by usig appriati ethds

2 I classical echaics, idetical particles ca be idetified by their psitis I quatu echaics, because f the ucertaity priciple, idetical particles are idistiguishable This effect is cected with the Pauli eclusi priciple ad is f ajr iprtace i deteriig the prperties f ats, uclei, ad bulk atter

3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ad te here we have used labels that idetify particlesad, tie idepedet Schrdiger equati is The Ψ, Ψ tie depedet Schrdiger equati is The Ψ the Ψ Csider tw particles,, E, V,t, V t,t, i,t,,t h h h h h

4 4 A iprtat case is whe the particles d t iteract with each ther This is called the idepedet particle del It s the startig pit fr the He at e.g. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t Which leads ad, we ca write This suggests,,, The E V d d E V d d E E E E V V V V V h h h h

5 et s say particle is i quatu state ad particle is i quatu state ( ) ( ) ( ), If we iterchage the tw particles the ( ) ( ) ( ), We get differet wave fuctis (ad prbability desities) s the tw particles are distiguishable Tw particles are idistiguishable if we ca echage their labels withut chagig a easurable quatity Thus either f the slutis abve is satisfactry 5

6 I fact there are tw ways t cstruct idistiguishable wave fuctis ( ( ) ( ) ( ) ( )) s a Ψ s is a syetric wave fucti uder particle iterchage f Ψ s Ψ s ( ( ) ( ) ( ) ( )) Ψ a is a atisyetric wave fucti uder particle iterchage f Ψ a -Ψ a The prbability desity reais uchaged uder particle iterchage f 6

7 All particles with iteger spi are called bss Spi 0,,, Pht, pis, Z-bs, Higgs All particles with half iteger spi are called feris Spi /, /, Electr, prt, eutr, quarks, Fr the et few lectures we ll fcus the feris (electrs) 7

8 The wave fucti f a ulti-particle syste f idetical feris is atisyetric uder iterchage f ay tw feris f idetical bss is syetric uder iterchage f ay tw bss The Pauli eclusi priciple fllws fr these priciples N tw idetical electrs (feris) ca ccupy the sae quatu state a fr (sae quatu state) a ( ( ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( )) 0 8

9 9 What abut particles? Syetric Atisyetric Use the Slater deteriat ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) perutatis s s ( ) ( ) ( ) () ( ) ( ) () ( ) () () ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () () () () () ()) ( 6 det! a a

10 Just a reider, we are presetly ly wrkig with the space wave fuctis We ll get t spi i a little bit A csequece f idetical particles is called echage frces Syetric space wave fuctis behave as if the particles attract e ather Atisyetric wave fuctis behave as if the particles repel e ather 0

11 Ifiite Square Well Quick review. Fr a ifiite well at 0 ad Slutis are () E π h si π

12 et s retur t the ifiite well prble ly w with tw particles Case distiguishable particles ( ) ( ) ( ), Case idetical particles (syetric) (, ) ( ( ) ( ) ( ) ( )) Case idetical particles (atisyetric) (, ) ( ( ) ( ) ( ) ( ))

13 What is the prbability that bth particles will be the left side f the well? Case ( ) / 0 / 0, d d P ( ) ( ) si si / 0 / 0 / 0 / 0 P d d P d d P π π

14 4 Case ad Fr a syetric wave fucti () the particles are re likely the sae side (attracted) Fr the atisyetric wave fucti (-) the particles are re likely ppsite sides (repel) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) 0 where / 0 / 0 / 0 / 0 > ± ± ± ± K K P d P d d P d d P

15 The wave fucti f a ulti-particle syste f idetical feris is atisyetric uder iterchage f ay tw feris f idetical bss is syetric uder iterchage f ay tw bss The Pauli eclusi priciple fllws fr these priciples N tw idetical electrs (feris) ca ccupy the sae quatu state a fr (sae quatu state) a ( ( ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( )) 0 5

16 Peridic Table The Pauli eclusi priciple is the basis f the peridic table This is why all the electr s d t siply fall it the grud state because they are feris Recall there were 4 quatu ubers that specified the cplete hydrge wave fucti:, l, l, ad s N tw electrs i the sae at ca have these sae quatu ubers 6

17 Peridic Table Buildig the peridic table Priciple quatu uber frs shells Orbital agular etu quatu uber l frs subshells l0,,,, are called s,p,d,f, Each l ca hld tw electrs, e spi up ( s /) ad e spi dw ( s -/) The electrs ted t ccupy the lwest eergy level pssible Electrs bey the Pauli eclusi priciple 7

18 Eergy levels Peridic Table Nte fr ultielectr ats, states f the sae ad differet l are lger degeerate This is because f screeig effects Referrig back t the radial prbability distributis, because the s states have -zer prbability f beig clse t the ucleus, their Culb ptetial eergy is lwer 8

19 Iizati Ptetial 9

20 Hydrge (H) s Heliu (He) Peridic Table s clsed shell ad cheically iert ithiu (i) s s valece, lw I, partially screeed, cheically very active Berylliu (Be) s s Clsed subshell but the s electrs ca eted far fr the ucleus 0

21 Peridic Table Br (B) s s p Saller I tha Be because f screeig Carb (C) s s p I actually icreases because the electrs ca spread ut i / l state lbes The valece is 4 sice a eergetically favrable cfigurati is s s p Nitrge (N) s s p See cets fr C. Electrs spread ut i / l state lbes Oyge (O) s s p 4 Tw f the l state electrs are paired Electr-electr repulsi lwers I

22 Flurie (F) Peridic Table s s p 5 Very cheically active because it ca accept a electr t bece a clsed shell Ne (Ne) s s p 6 ike He

23 Peridic Table

24 Peridic Table Peridic table is arraged it grups ad perids Grups Have siilar shell structure hece have siilar cheical ad physical prperties Eaples are alkalis, alkali earths, halges, iert gases Perids Crrespd t fillig d ad f subshells Eaples are trasiti etals (d,4d,5d), lathaide (4f), ad actiide (5f) series Because there are ay upaired electrs, spi is iprtat fr these eleets ad there are large agetic effects 4