QMGreensite_merged

318 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

WearenowreadytoexpressanylinearoperatorMintermsof|u><v|symbols.
Wehave

M = IMI

=

(

∑

i

|ei><ei|

)

M





∑

j

|ej><ej|





=

∑

ij

|ei><ei|M|ej><ej|

=

∑

ij

mij|ei><ej| (21.92)

Exercise: Let

|v′> = M|v>

<u′| = <u|M (21.93)

wherewedenotethecomponentsofthebra-ketvectorsshownas

|v> =

∑

i

vi|ei>

|v′> =

∑

i

v′i|ei>

<u| =

∑

i

ui<ei|

<u′| =

∑

i

u′i<ei| (21.94)

Usingthebra-ketrepresentationofMineq. (21.92),showthatthecomponentsvi→
v′ibymatrixmultiplicationofacolumnvector,andui→u′ibymatrixmultiplication
ofarowvector.

Exercise: Let{en}and{e′m}betwodifferentbasesforthe samevectorspace.
ShowthattheD×Dmatrixwhoseelementsare

Uij=<ei|e′j> (21.95)

isaunitarymatrix.

Ingeneral,thebravectorwhichcorrespondstoM|v>isnotthesameas<v|M,
i.e.
<Mv|+=<v|M (21.96)