QMGreensite_merged

21.2. LINEARALGEBRAINBRA-KETNOTATION 317

LikewiseLactsonbravectors<q|bytakinganinnerproductontheright

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

= <q|u><v| (21.84)

Wenowshowthatanylinearoperatorcanbeexpressedasalinearcombination
of|u><v|symbols. BeginwiththeidentityoperatorI,definedsuchthat

I|v>=|v> and <u|I=<u| (21.85)

forall<u|, |v>inthevectorspace. Itiseasytoseethatsuchanoperatorcanbe
written,inagivenbasis{|ei>}as

I=

∑

n

|en><en| (21.86)

Checkthis:

I|v> =

(

∑

n

|en><en|

)(

∑

k

vk|ek>

)

=

∑

n

∑

k

vk|en>δnk

=

∑

k

vk|ek>

= |v> (21.87)

Likewise

<u|I =

(

∑

k

uk<ek|

)(

∑

n

|en><en|

)

=

∑

k

∑

n

ukδkn<en|

=

∑

k

uk<ek|

= <u| (21.88)

Finally,ifMisanylinearoperator

IM=MI=M (21.89)

because
IM|v>=I|Mv>=|Mv>=M|v> (21.90)

and
MI|v>=M|v> (21.91)