21.3. HILBERTSPACE 329
= [φ∗ 0 (x),φ∗ 1 (x),φ∗ 2 (x),...]
φ 0 (y)
φ 1 (y)
φ 2 (y)
.
.
.
(21.160)
Takingtheinnerproductoftherowandcolumnvectors,wegettheidentity
∑∞
n=0
φ∗n(x)φn(y)=δ(x−y) (21.161)
Nextweturntothematrixelementsofoperators. Takingtheinnerproductof
theeigenvalueequationHho|φn>=En|φn>withthebra<φm|gives
Hmnho ≡<φm|Hho|φn>=Enδmn (21.162)
i.e.Hmnho isadiagonalmatrix
Hho=
E 1 0 0...
0 E 2 0...
0 0 E 3...
......
......
......
(21.163)
ThematrixrepresentationsXmnandPmnareeasilyobtainedusingtheraisingand
loweringoperatorrelations
X =
√
̄h
2 mω
(a†+a)
P = i
√
mω ̄h
2
(a†−a)
a†|φn> =
√
n+ 1 |φn+1>
a|φn> =
√
n|φn− 1 > (21.164)
Then
Xmn =
√
̄h
2 mω
[
<φm|a†|φn>+<φm|a|φm>
]
=
√
̄h
2 mω
[√
n+ 1 δm,n+1+
√
nδm,n− 1
]
(21.165)
andlikewise
Pmn=i
√
mω ̄h
2
[√
n+ 1 δm,n+1−
√
nδm,n− 1
]
(21.166)