a$)$ The CGC possess the symmetry property

hl$1,$ l$2$; m$1,$ m$2|$l$,$ mi $=(1)$l$$l$1$l$2$

hl$2,$ l$1$; m$2,$ m$1|$l$,$ mi$.(1)$

Use this property to show that for two electrons in equivalent orbitals the total orbital

angular momentum quantum number l must have even values for singlet $($spin$)$ states and

odd values for triplet states.

b$)$ Prove the recurrence relation

hl$1,$ l$2$; m$11,$ m$2+1|$l$,$ li $=$

s

l$2($l$2+1)$ m$2($m$2+1)$

l$1($l$1+1)$ m$1($m$11)$hl$1,$ l$2$; m$1,$ m$2|$l$,$ li $(2)$

c$)$ Verify this recurrence relation explicitly for the coefficient hl$1,$ l$2$; m$1,$ m$2|$l$,$ li $=$ h$1,$

$1$

$2$

; $1,$

$1$

$2$

$|$

$1$

$2$

$,$

$1$

$2$

i

using the table on the next page.