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)