QMGreensite_merged

310 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

or,inthecaseofaD×Dmatrix,solvetheDsimultaneousequations

∑D

j=1

miju(jn)=λnu(in) (i= 1 , 2 , 3 ,...,D) (21.34)

Thegeneralsolutionofthisequation,intermsofdeterminants,isgiveninanylinear
algebrabook.Thesolutionisdenotedu(n),andshouldbenormalizedsothat

[u∗ 1 ,u∗ 2 ]

[

u 1

u 2

]

=

∑D

i=1

u∗iui= 1 (21.35)

Theorem

Theeigenvalues ofaHermitianmatrixareallreal. Anytwoeigenvectorsof a
Hermitianmatrix,whichcorrespondtodifferenteigenvalues,areorthogonal.

Proof: Let u(m) andu(n) betwoeigenvectorsof a Hermitianmatrix H, with
eigenvaluesλn, λmrespectively. Considerthequantity

Q=

∑

ij

(uni)∗Hiju(jm) (21.36)

Sinceumisaneigenvector,weknowthat
∑

j

Hiju(jm)=λmu(im) (21.37)

so
Q=λm

∑

i

(u(in))∗u(im)=λmu(n)·u(m) (21.38)

Ontheotherhand,wecanwriteQas

Q=

∑

ij

(

Hij∗u(in)

)∗

u(jm) (21.39)

BecauseHisanHermitianmatrix,

Hij∗=Hji (21.40)

so

Q =

∑

ij

(

Hjiu(in)

)∗

u(jm)

=

∑

j

(λnu(jn))∗u(jm)

= λ∗nu(n)·u(m) (21.41)