Lecture8 - Helium(Z=2 Lets move on to atoms with more than...

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Helium’(Z=2)’ Let±s’move’on’to’atoms’with’more’than’one’electron:’He’ H = ! ! 2 2 m " r 1 2 ! ! 2 2 m " r 2 2 ! 2 e 2 4 # ! 0 r 1 ! 2 e 2 4 ! 0 r 2 + e 2 4 ! 0 | r 2 ! r 1 | kine;c’terms’ e1-nucleus’ e2-nucleus’ e-e’repulsion’ ignore’for’now’ Ignoring’the’interac;on’term,’we’can’solve’the’Schrodinger’equa;on’exactly.’ The’result’is’two’ hydrogenic’solu;ons’for’each’electron,’only’with’Z=2.’ ! ( r 1 , r 2 ) ± = 1 2 nlm ( r 1 ) " n " l " m ( r 2 ) ± n ' l ' m ' ( r 1 ) nlm ( r 2 ) [ ] Which’is’either’spa;ally’symmetric’(+)’called’the’ para state’or’an;symmetric’(-),’the’ ortho
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Ground’state’energy’of’He’ For’the’ground’state,’both’electrons’are’in’the’( n=1,l=0,m=0)’( ortho’vanishes)’ E 0 = ! 13.6 Z 2 eV So’the’ground’state’of’the’He’atom’ignoring’ e-e’repulsion:’ E 0 = ! 13.6 " (4) ! 13.6 " 4 = ! 108.8 eV E = ! Z 2 E 0 n 1 2 ! Z 2 E 0 n 2 2 And’so’the’addi;on’of’the’two’single’par;cle’energies:’ The’actual’ground’state’energy’is’-78.975’ eV.’ e-e’repulsion’is’sizeable’correc;on’ Straight’applica;on’of’perturba;on’theory’is’inappropriate,’since’it±s’a’large’contribu;on.’ Recall’from’Lecture’2,’the’ground’state’energy’of’the’Bohr’H’atom’was:’ !
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Lecture8 - Helium(Z=2 Lets move on to atoms with more than...

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