Mathematical Methods for Physicists: A concise introduction - Site Map

SOME SPECIAL OPERATORS
jui ˆ Xn
jˆ1
u j ji; j u j ˆ hjui: j
…5:27†
We may write the above as
jui ˆ
Xn
jˆ1
jij j hj
!
jui;
which is true for all jui. Thus the object in the brackets must be identi®ed with the
identity operator:
I
~
ˆ Xn
jˆ1
jij j hjˆ Xn
jˆ1
P
~
j:
…5:28†
Now we will see that the e€ect of this particular projection operator on jui is to
produce a new vector whose direction is along the basis vector j ji and whose
magnitude is h jjui:
P jjui ˆ jij j hjui ˆ jiu j j :
~
We see that whatever jui is, P jjui is a multiple of j ji with a coecient u j which is
~
the component of jui along j ji. Eq. (5.28) says that the sum of the projections of a
vector along all the n directions equals the vector itself.
When P j ˆ jij j hjacts on j ji, it reproduces that vector. On the other hand, since
~
the other basis vectors are orthogonal to j ji, a projection operation on any one of
them gives zero (null vector). The basis vectors are therefore eigenvectors of P k ~
with the property
P kjiˆ j kj ji; j
~
… j; k ˆ 1; ...; n†:
In this orthonormal basis the projection operators have the matrix form
0
1 0
1 0
1
1 0 0
0 0 0
0 0 0
0 0 0
P 1 ˆ
~
B
@
0 0 0 C
. . . A ; 0 1 0
0 0 0
P ~ 2 ˆ
B
@
0 0 0 C
. . . .
. . A ; P ~ N ˆ
0 0 0
:
B
C
.
@
. .
.
.
. . .
.
A
.
1
Projection operators can also act on bras in the same way:
hujP j ˆ hujjih jj ˆ u* j hj: j
~
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