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316 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

Sinceboth|v′>andM|ei>areketvectors,theymusthaveanexpansioninbasis
vectorswhichwewriteas
|v′>=

∑

k

v′k|ek> (21.77)

and
M|ej>=

∑

k

mkj|ek> (21.78)

Substitutinginto(21.76)gives
∑

k

v′k|ek>=

∑

i

∑

k

vimki|ek> (21.79)

Takingtheinnerproductofbothsideswiththebravector<ei|,andusingagainthe
bi-linearitypropertyandorthonormalityproperties(21.62),(21.63)

<ei|

∑

k

v′k|ek> = <ei|

∑

j

∑

k

vjmkj|ek>

∑

k

vk′ <ei|ek> =

∑

j

∑

k

vjmkj<ei|ek>

∑

k

vk′δik =

∑

j

∑

k

vjmkjδik

v′i =

∑

j

mijvj (21.80)

Butthisisjusttherule,incomponents,formultiplyingacolumnvectorbyamatrix.
Thismeansthat,inagivenbasis,theactionofalinearoperatoronvectorsisequiv-
alenttomatrixmultiplication. Infact,takingtheinnerproductofbothsidesofeq.
(21.78)withthebravector<ei|,wefind

TheMatrixElementofOperatorM inthebasis{|en>}

mij=<ei|M|ej > (21.81)

Next,itis useful to introduce alinearoperationLon brasandkets whichis
representedsymbollicallyby
L=|u><v| (21.82)

ThemeaningofthissymbolisthatLoperatingonany ket|w >willturnitinto
anotherket,proportionalto|u>,bytakinganinnerproductontheleft

L|w> = (|u><v|)|w>=|u>(<v|w>)

= (<v|w>)|u> (21.83)