9.2. ALGEBRAANDEXPECTATIONVALUES 151
andwecanalsoderiveasimilarequationforaφn:
a|φn> =
1
√
n!
a(a†)n|φ 0 >
=
1
√
n!
[(a†)na+n(a†)n−^1 ]|φ 0 >
= n
1
√
n!
(a†)n−^1 |φ 0 >
=
√
n
1
√
(n−1)!
(a†)n−^1 |φ 0 >
=
√
n|φn− 1 > (9.67)
Insummary:
a|φn> =
√
n|φn− 1 >
a†|φn> =
√
n+ 1 |φn+1>
x =
√√
√√ ̄h
2 mω
(a+a†)
p =
1
i
√√
√√mωh ̄
2
(a−a†) (9.68)
Asanexampleoftheuseoftheserelations,letuscomputethepositionuncertainty
∆xinthen-thenergyeigenstate.Asusual
(∆x)^2 =<x^2 >−<x>^2 (9.69)
Now
√
2 mω
̄h
<x> = <φn|(a+a†)|φn>
=
√
n<φn|φn− 1 >+
√
n+ 1 <φn|φn+1>
= 0 (9.70)
Next,applyingsuccessivelytherelations(9.68)
2 mω
̄h
<x^2 > = <φn|(a+a†)(a+a†)|φn>
= <φn|(aa+aa†+a†a+a†a†)|φn>
= <φn|aa|φn>+<φn|aa†|φn>+<φn|a†a|φn>+<φn|a†a†|φn>
=
√
n<φn|a|φn− 1 >+
√
n+ 1 <φn|a|φn+1>