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)