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=0φ∗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)