210 CHAPTER13. ELECTRONSPIN
Thisisthecolumnvectorrepresentationofthevector%v,inthebasis{%e 1 , %e 2 }. You
willnoticethatthecomponentsofthecolumnvectorareinnerproductsoftheket
vector%vwiththebravectors{%e 1 ,%e 2 },i.e.
a = %e 1 ·%v=<e 1 |v>
b = %e 2 ·%v=<e 2 |v> (13.29)
Wecanchoose%v=%e 1 or%e 2 ,andfindthevaluesofaandbforthebasisvectors. Its
easytoseethatthecorrespondingcolumnvectorsare
%e 1 ↔
[
1
0
]
%e 2 ↔
[
0
1
]
(13.30)
InLecture 3 wediscussedthenotionof alinearoperator: itssimply arule for
changinganyvector%vintoanothervector%v′
v%′=M%v (13.31)
suchthat,foranytwovectors,
M(a%u+b%v)=aM%u+bM%v (13.32)
ThematrixelementmijofalinearoperatorM inthebasis{%ek}isgivenbythe
innerproduct
Mij=%ei·M%ej (13.33)
or,inbra-ketnotation
Mij=<ei|M|ej> (13.34)
Supposewearegiventhematrixelements ofsome linearoperatorM inacertain
basis.Thenitseasytoseethatthecomponents
v′i=<ei|v′> (13.35)
ofthetransformedvector|v′>=M|v>,intermsofthecomponents
vi=<ei|v> (13.36)
oftheoriginalvectoraregivenbytheusualruleofmatrixmultiplication
vi′ = <ei|v′>
= <ei|M|v>
= <ei|M
∑
j
vj|ej>
=
∑
j
<ei|M|ej>vj
=
∑
j
Mijvj (13.37)