QMGreensite_merged

21.2. LINEARALGEBRAINBRA-KETNOTATION 315

Ofcourse,incallingthecomponentsofavectora”wavefunction”,weareantici-
patingtheuseofeq. (21.69)inquantummechanics.

Next,wewanttoknowthecomponentsofthebra-vector<v|correspondingto
|v>.Writing
<v|=

∑

m

cm<em| (21.70)

taking theinner productwith |en >, and againusing thebilinearity (21.62)and
orthonormality(21.63)properties,wefind

cn=<v|en> (21.71)

Then,usingthefactthat<u|v>=<v|u>∗(eq.(21.61)),andeq. (21.69)

cn=vn∗ (21.72)

Therefore,inagivenorthonormalbasis{en},thecorrespondingbraandketvectors
havetheform

|v> =

∑

n

vn|en>

<v| =

∑

n

v∗n<en| (21.73)

Thefact thatthe componentsof thebravectorarethe complexconjugateofthe
componentsoftheketvectorisinagreementwithwhatwasalreadybeenstatedin
equations(21.58)and(21.59).

• LinearOperators Justasafunctionf isarulefortakinganynumber(x)
  andturningitintoanothernumbery(i.e. y=f(x)),soananoperatorMisarule
  fortakinganyvector|v>intoturningitintosomeothervector|v′>,

|v>→|v′>=M|v> or |Mv> (21.74)

ALinearOperatorhastheproperty

M[a|v>+b|u>]=aM|v>+bM|u> (21.75)

forany vectors |u>, |v >andconstantsa, b. Becauseof thisproperty, wecan
determinewhatalinearoperatorM doestoanyvector|v>byspecifyingwhatit
doestoanybasisvector|en>:

|v′>=M|v> = M

∑

j

vj|ej>

=

∑

j

vjM|ej> (21.76)