QuantumPhysics.dvi

(ii) Next, leta′ 6 =abe two distinct eigenvalues (which are both real by (i)),

A|φ〉 = a|φ〉

A|φ′〉 = a′|φ′〉 (3.40)

Taking the inner product of the first line with〈φ′|and of the second line by〈φ|, and using

〈φ|φ′〉=〈φ′|φ〉∗, and〈φ|A|φ′〉=〈φ′|A†|φ〉∗=〈φ′|A|φ〉, we find that

〈φ′|A|φ〉=a〈φ|φ′〉=a′〈φ|φ′〉 (3.41)

Sincea′ 6 =a, we must have〈φ′|φ〉= 0 which proves (ii).

Constructing eigenvalues and eigenvectors in general is difficult, even in finite dimension.

IfAis anN×Nmatrix, the eigenvalues obey thecharacteristic equation,

det(aI−A) =aN+c 1 aN−^1 +c 2 aN−^2 +···+cN= 0 (3.42)

wherec 1 =−trAandcN= (−)NdetA. Clearly, for HermitianA, all coefficientscnare real,

and all roots are real. Assuming that, givenA, the rootsa 1 ,a 2 ,···,aN of this algebraic

equation have been found (possibly numerically), then finding the associated eigenvectors

reduces to a linear problem,

(anI−A)|φ〉= 0 (3.43)

which can be solved by standard methods of matrix algebra.

(iii) Finally, a given eigenvalueai, may have one or several linearly independent eigenvectors,

which span the entire eigenspaceEiassociated withai. By the result of (ii), the eigenspaces

associated with distinct eigenvalues are also mutually orthogonal. Therefore, the entire

Hermitian matrix equals,

A=

∑

i

aiPi ai 6 =aj when i 6 =j (3.44)

where Pi represents the projection operator on eigenspace Ei. The dimension dimPi is

referred to as thedegeneracy (or multiplicity) of the eigenvalueai.This number clearly co-

incides with the degeneracy of the rootaiin the characteristic equation. In matrix notation,

this produces a block-diagonal representation ofA,

A=











a 1 I 1 0 0 ··· 0

0 a 2 I 2 0 ··· 0

0 0 a 3 I 3 ··· 0

···

0 0 0 ··· amIm











(3.45)

This proves (iii).