QMGreensite_merged

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)