tonic.physics.sunysb.edutonic.physics.sunysb.edu/~dteaney/f17_phy503/lectures/hw8_sol.pdf · we...
TRANSCRIPT
We recall Chat the action of a group in a function is given
b9Ogfcx) = fcgtx )
⇒ ( Or, f. Or
,
h ) = |d2x f*CF' x ) h ( 9-'
x )
In this case the representation for the inverse of a
rotation in the coordinate space is given by
Doe's -
.
fs;sn¥±EMI)
Therefore,
we can do a transformation of coordinates in
the previous integralI
' =
Dcri'
) e
⇒ ( Or, f. Or ,h > =) dx
'
wet JI f*CE ' )htx ' )
but in tn 's case kktJl=ldetEssn¥¥IIs¥¥H=L
⇒ ( Or .f,
Or ,h)=)d2x'
f*tx' )htx
'
) = ( fin )
Since the inner product is invariant under representationsof the group
we have
( fin'
, to'
)=tµ§ loofah,
ogfi"
) -
to.
§ with sacks# Doha
= §.
C fin'
.fi'
)took
( Diana ) )*DYits)= saw , too'
) daddabdm
= § ( fin'
, fin'
) dabdnu - Ccmdmdab
where on'
'
-
{ ( fin'
, fin'
>
By looking at the slides,
we see thatany function can
be decomposed in a basis determined by the differentrepresentations and the rows of each representation .
In this
sense,
we can think of the fin'
as the projection ofsaid function f in this basis
.
Since in this group we know that the inner product is
invariant under the group operations we can see that
( fin'
,Hfbi
"
t.nl.
§ last 'a"
,9Hfi"
) =tµ§( osfi'
, Hosts" )
Where we used the commutation of H withg
in the last
inequality .Now
, using the results from b) we see that
Ogfan'
- fin'
DTACS ) Osf 's'
= fed"
Dmdb (9)
⇒ ( fam,
ntfi"
) =
t.gs#',9tfoiYEalDYatsD*Dmdbc9) )
= E ( fin, Hfoi
"
) dad Sabdm = § hf in
, Hfi'
) dabdm:' n'
dabdnv
where we expresshen '
-
§ 1 fin'
, ntfin'
).