QMGreensite_merged

21.3. HILBERTSPACE 321

Inparticular,thematrixelementsofM intheM-basisformadiagonalmatrix

Mmn = <vm|M|vn>

= λnδmn (21.109)

andtherefore

M =

∑

mn

λnδmn|vm><vn|

=

∑

n

λn|vn><vn| (21.110)

For thisreason, solving the eigenvalueequation M|vn >= λn|vn > issometimes
referedtoas ”diagonalizing”the operatorM, sincethematrixMmnisadiagonal
matrixinthebasisspannedbythetheeigenvectors{|vn>}

21.3 Hilbert Space

Hilbert spaceis aninfinite-dimensional, linearvector space. The eigenvectors (or
eigenstates)ofanyHermitianoperator,havinganinfinitenumberofnon-degenerate
eigenvalues,formabasisinHilbertspace.
SofarwehaveencounteredtheeigenvalueequationforHilbertspaceoperatorsin
threeversions.First,thereisthe”operator-acting-on-a-function”version

O ̃ψn(x)=λnψn(x) (21.111)

Second,the”matrix-multiplication”versionis

∫

dyO(x,y)ψn(y)=λnψ(x) (21.112)

wherethematrixelementO(x,y)wasdefinedbytheoperatorOactingonadelta-
function
O(x,y)=O ̃δ(x−y) (21.113)

Finally, there isthe abstract”bra-ket”version, whichholdsjust as wellin finite
dimensionallinearvectorspaces

O|ψn>=λn|ψn> (21.114)

Weask: whatistherelationshipofthefirst twoformsof theeigenvalueequation
withthethird”bra-ket”form,and,inparticular, whatistherelationshipbetween
theketvector|ψ>andthewavefunctionψ(x)?Theanswer,briefly,isthatthefirst
twoformsaretheeigenvalueequationinthex-representation,andthewavefunction
ψ(x)givesthecomponentsofthestatevector|ψ>inthex-representation.