21.3. HILBERTSPACE 331
Theselookjust like the eigenstates of the harmonicoscillator Hamiltonian φn in
theHO-representation,but ofcoursetheycorrespondtoverydifferentstates. An
eigenfunctionofposition|x>,inthesquare-wellrepresentation,is
ψx(n) = <φn|x>
= <x|φn>∗
= φn(x)=√
2
Lsin(
nπx
L)
(21.171)Asacolumnvector,aneigenstateofpositionhastheform
ψx=√
2
L
sin(
πx
L)sin(
2 πx
L)sin(
3 πx
L).
.
.
(21.172)
Theorthogonalityofpositioneigenstatesimplies
δ(x−y) = <x|y>= [φ∗ 1 (x),φ∗ 2 (x),φ∗ 3 (x),...]
φ 1 (y)
φ 2 (y)
φ 3 (y)
.
.
.
(21.173)
Takingtheinnerproductof therow andcolumnvectors,wegetanotheridentity,
analogousto(21.161)
2
L∑∞
n=1sin(nπxL)
sin(nπyL)
=δ(x−y) (21.174)Asinthecaseoftheharmonicoscillator,thematrixelementsofthesquare-well
Hamiltonianinthesquare-wellrepresentationisdiagonal:
Hmnsq ≡<φm|Hsq|φn>=Enδmn (21.175)i.e.,asamatrix
Hsq=
E 1 0 0...
0 E 2 0...
0 0 E 3...
......
......
......