QMGreensite_merged

150 CHAPTER9. THEHARMONICOSCILLATOR

Rescalingx

y=

√

mω

̄h

x (9.62)

Thegeneralsolutionbecomes

φ(x) =

1

√

n!

(mω

π ̄h

) 1 / 4 ( 1

√

2

)n(

y−

∂

∂y

)n

e−y

(^2) / 2

=

1

√

n!

(

mω

π ̄h

) 1 / 4 (

1

√

2

)n

Hn(y)e−y

(^2) / 2
(9.63)
withcorrespondingeigenvalues
En= ̄hω(n+

1

2

) (9.64)

ThefunctionsHn(y)areknownasHermite Polynomials. Operatingntimeson
thegroundstatewiththeraisingoperatora†resultsinafunctionwhichisjustthe

groundstatemultipliedbyann-thorderpolynomialinthevariabley=

√
mω/h ̄x.
ThesearetheHermitepolynomials.ThefirstseveralHermitepolynomialsare:

H 0 (y) = 1

H 1 (y) = 2 y

H 2 (y) = 4 y^2 − 2

H 3 (y) = 8 y^3 − 12 y (9.65)

Byapplyingeq. (9.63),theHermitepolynomials,andtheeigenstatesφn(x),canbe
determinedtoanyorderdesired.

9.2 Algebra and Expectation Values

Oneofthemostremarkablepropertiesofthe quantumharmonicoscillatoristhat
manycalculationsinvolvingexpectationvaluescanbedonealgebraically,withoutever
usingtheexplicitformoftheeigenstatesφn(x).Thereasonsforthisare,first,thatthe
xandpoperatorsarelinearcombinationsoftheraisingandloweringoperators(eq.
(9.27));andsecond,thattheraisingandloweringoperatorshaveasimplealgebraic
actionontheeigenstates.
Considertheraisingoperatoractingonthestateφn. Usingeq. (9.61)wehave

a†|φn> =

1

√

n!

(a†)n+1|φ 0 >

=

√

(n+1)

1

√

(n+1)!

(a†)n+1|φn+1>

=

√

n+ 1 |φn+1> (9.66)