21.3. HILBERTSPACE 323
variousangular-momentumrepresentations. Theharmonic-oscillatorrepresentation
is frequently usedinquantum field theory, andthe angular-momentum represen-
tationswillbeemployedshortly, inourdiscussionofspinandadditionofangular
momentum.
Supposewe have an HermitianoperatorO withaset of eigenstates andnon-
degenerateeigenvaluessatisying
O|φn>=on|φn> (21.123)Thenthewavefunctionofanystate|ψ>intheO-representationistheinnerproduct
ψ(n)=<φn|ψ> (21.124)andthematrixelementofanylinearoperatorMintheO-representationisgivenby
MijO=<φi|M|φj> (21.125)Wewillconsiderasexamplesfouroperators: positionX,X|x 0 >=x 0 |x 0 > x 0 ∈[−∞,∞] (21.126)momentumP,
P|p 0 >=p 0 |p 0 > p 0 ∈[−∞,∞] (21.127)
theharmonicoscillatorHamiltonianHho,
Hho|φn>=En|φn> En= ̄hω(n+1
2
) (21.128)
andthesquarewellHamiltonianHsq,
Hsq|φi>=Ei|φi> Ei=n^2̄h^2
2 mL^2(21.129)
FromthematrixelementsandeigenfunctionsoftheseoperatorsintheX-representation,
wecanconstructthematrixelementsandeigenstatesinotherrepresentations. All
ourworksofarhasbeenintheX-representation,sowebeginthere.
- TheX-representation
TakingtheinnerproductoftheeigenvalueequationX|y>=y|y>withthebra
<x|,wegetimmediatelythematrixelements
<x|X|y> = y<x|y>
= yδ(x−y)
= xδ(x−y) (21.130)