QMGreensite_merged

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