176 CHAPTER11. ANGULARMOMENTUM
wehave
L ̃^2 = L ̃+L ̃−− ̄hL ̃z+L ̃^2 z
= L ̃−L ̃++[L ̃+,L ̃−]− ̄hL ̃z+L ̃^2 z
= L ̃−L ̃++h ̄L ̃z+L ̃^2 z (11.26)Finally,weneedthecommutatorsofL ̃zwithL ̃±:
[L ̃z,L ̃+] = [L ̃z,L ̃x]+i[L ̃z,L ̃y]
= i ̄hL ̃y+i(−i ̄hL ̃x)
= h ̄L ̃+ (11.27)andlikewise
[L ̃z,L ̃−]=− ̄hL ̃− (11.28)
Insummary,thecommutatorsof(11.14)intermsofL ̃x, L ̃y, L ̃zcanbereplaced
bycommutatorsinvolvingL ̃±,L ̃z,andL ̃^2 canalsobeexpressedintermsofL ̃±, L ̃z:
[L ̃+,L ̃−] = 2 ̄hL ̃z
[
L ̃z,L ̃±]
= ± ̄hL ̃±
L ̃^2 = L ̃+L ̃−+L ̃^2 z− ̄hL ̃z
= L ̃−L ̃++L ̃^2 z+ ̄hL ̃z (11.29)ThepointofintroducingtheL ̃±operatorsisthattheyactoneigenstatesofL ̃z
justliketheraisingandloweringoperatorsaanda†actoneigenstatesoftheharmonic
oscillatorHamiltonian.Let
φ′=L+φab (11.30)
Then
Lzφ′ = LzL+φab
= (L+Lz+ ̄hL+)φab
= ( ̄hb+ ̄h)L+φab
= h ̄(b+1)φ′ (11.31)andweseethatφ′=L+φabisalsoaneigenstateofLz,withtheeigenvalue ̄h(b+1).
TheraisingoperatorhasthereforeraisedtheeigenvalueofLzbytheamount ̄h. The
”raised”eigenstateφisalsoaneigenstateofL^2 ,withthesameeigenvalueasφab:
L ̃^2 φ′ = L ̃^2 L ̃+φab
= L ̃+L ̃^2 φab
= ̄h^2 a^2 L ̃+φab
= ̄h^2 a^2 φ′ (11.32)