9.1. RAISINGANDLOWERINGOPERATORS 143
Thenthereisasimpleidentity
a^2 +b^2 =c∗c (9.10)
Letsfindthecorrespondingidentityforoperators. Insteadoftworealnumbers,con-
sidertwoHermitianoperatorsAandB,which,ofcourse,haveonlyrealeigenvalues.
Definethenon-Hermitianoperator
C=A+iB (9.11)
whosehermitianconjugateis
C†=A−iB (9.12)
ThenthesumofsquaresofAandBsatisfiestheidentity
A^2 +B^2 =C†C−i[A,B] (9.13)
andthecommutatorofCwithC†is
[
C,C†
]
=− 2 i[A,B] (9.14)
Theserelationsareeasytocheck(exercise!),keepinginmindthatoperatorsAand
Bdonotnecessarilycommute. Now,theHamiltonianoperatorcanbewrittenasa
sumofsquaresofHermitianoperators
H ̃=
(
p ̃
√
2 m
) 2
+(
√
V(x))^2 (9.15)
so,writing
A =
√
V
B =
p ̃
√
2 m
C =
√
V+i
p ̃
√
2 m
(9.16)
wehave
H ̃=C†C−i
√
1
2 m
[
√
V,p ̃] (9.17)
and
[
C,C†
]
=− 2 i
√
1
2 m
[
√
V,p ̃] (9.18)
ThisisnotaparticularlyusefulwaytoexpressH ̃ inmostsituations;generallythe
commutator[
√
V,p ̃]isjustanothercomplicatedoperator.Inthecaseoftheharmonic