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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>