312 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
so
λ 1 =+1 u^1 =1
√
2
[
1
−i]
(21.51)Theprocedureforλ 2 =− 1 isidentical:
[
0 i
−i 0][
u^21
u^22]
= −[
u^21
u^22][
iu^22
−iu^21]
=[
−u^21
−u^22]
(21.52)Andthistimeu^22 =iu^21 .Normalizingtodetermineu^21 ,wefind
λ 2 =− 1 u^2 =1
√
2
[
1
i]
(21.53)Notethattheinnerproduct
u^1 ·u^2 =1
2
[1,i][
1
i]
= 1 +i^2 = 0 (21.54)vanishes,sou^1 andu^2 areorthogonal,againaspredictedbytheorem.
21.2 Linear Algebra in Bra-Ket notation
Thedefinitionofacolumnvectoras asetofnumbers isclearlynotadequateasa
definitionoftheword”vector”inphysics. Take,forexample,theelectric fieldE,
whichisathree-dimensional vector quantity. Inaparticular setof xyz cartesian
coordinates,thevectorEmayberepresentedbythethreecomponents
Ex
Ey
Ez
(21.55)butthereisnothingsacredaboutanyparticularsetofcoordinates,andifwerotate
toanothercoordinateframe,thesamevectorEwillhavethreedifferentcomponents
Ex′′
E′y′
E′z′
(21.56)Inclassicalmechanicsandelectromagnetism,avectorquantityisjustdenoted,e.g.,
byaboldfacesymbolEorbyanarrowE%. Thisisthequantitywhich,inaparticular
referenceframe,isrepresentedbyaD-dimensionalcolumnvector.