21.2. LINEARALGEBRAINBRA-KETNOTATION 313
Laws ofmotionarefundamental,reference framesarenot, andoneof the ob-
jectivesof vector notationistoexpressthelawsofclassicalphysicsincoordinate-
independentform;e.g. Newton’sLawofmotion,andAmpere’sLaw
F%=m%a and ∇×B%=μ 0 %j (21.57)aretrueinanysystemofcartesiancoordinates. Vectornotation,withtheassociated
conceptsofgrad,div,andcurl,isanelegantandusefulwayofavoidingtheclutter
ofcoordinateindices,anddisplayingthecontentofdynamicalequationsinthemost
direct way. Inclassical physics, vectorsareusually three-(orinrelativity, four-)
dimensional,andtheircomponentsarereal-valued. Inquantumtheory,thevectorsof
interestareusuallyinfinite-dimensional,andtheircomponentsaretypicallycomplex-
valued. Bra-ket notation, introduced by Dirac, aims to accomplishfor quantum
theorywhatvectornotationdoesforclassicalmechanics;namely,toexpressthelaws
of motionasdirectlyas possible, inaway that isindependentof the (irrelevant)
referenceframe.
Becausethevectorsinquantummechanicscanhavecomplexcomponents,itis
importanttodistinguishbetweenvectorswhosecomponentsarecolumnvectors,and
vectorswhosecomponentsarerowvectors. Theformerarethe”ket”vectors|v>,
thelatterthe”bra”vectors<v|.Thereisaone-to-onecorrespondencebetweenbra
andketvectors.Ifthecomponents(insomereferenceframe)ofaketvectoraregiven
by
|v>↔
v 1
v 2
v 3
.
.
.
vD
(21.58)
thenthecomponentsofthecorrespondingbravectorare
<v|↔[v∗ 1 ,v∗ 2 ,v∗ 3 ,...,v∗D] (21.59)Thesymbol↔isusedtoremindusthat thevaluesof thecomponentsdependon
thechoiceofreferenceframe;theyareonlyaparticularrepresentationofthebra-ket
vectors.Thebra-ketvectorsthemselves,likeF%orE%,aremeaningfulindependentof
thebasischosen.
ALinearVectorSpaceisacollectionofvectors{|v>},whichiscompleteunder
vectoraddition,i.e.if|v^1 >and|v^2 >belongtothespace,sodoesthecombination
a|v^1 >+b|v^2 > (21.60)