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Note that the matrix representationof a certain linear operator on a vector
spacedepends onthe basis used,differentbasesgivesdifferentmatrix represen-
tations, and consequently the matrix elements given by (5.34) are different.
Also note that, as we are only considering operators mapping V to V, the
matrices representing the operators are n n matrices.
×
5.2.2 Outer products
Let x and y be two vectors in a vector space. By the outer product between
| i | i
thesevectorswemean x y . Thiscanbeconsideredtodefinealinearoperator
| ih |
on the vector space as is seen from the following formal calculation
(x y )z = x ( y z )= y z x . 5.35
| ih | | i | i h | i h | i| i
Thus the vector z is mapped to the vector µx where µ is the complex
| i | i
number y z .
h | i
The usefulness of this conceptbecomes clear when it is applied to orthonor-
mal basis vectors. Let i be an orthonormal basis for a vector space V. We
| i
have the expansion (5.19) of an arbitrary vector x
| i
105
n
x = α i
i
| i | i
Xi=1
and the equation (5.21) for the expansion coefficients
α = ix .
i
h | i
Then consider the operator
n
i i. (5.36)
| ih |
Xi=1
Letting it act on the vector x yields
| i
n n
( i i)x = i ix
| ih | | i | ih | i
Xi=1 Xi=1
n n
= ix i = α i = x .
i
h | i| i | i | i
Xi=1 Xi=1
But this equation is true for any vector x so we can identify the operator
| i
in equation (5.36) with the identity operator, or
n
i i =I. (5.37)
| ih |
Xi=1
This is known as the completeness relation.
The readermightworryabout the ambiguities inthe notationwhenwriting
expressions such as x y z . It is not clear whether this should be read as the
| ih | i
operator x y actingonthe state z orthenumber y z multiplyingthestate
| ih | | i h | i
x . However, there is no ambiguity and the expression can be read in either
| i
way. It denotes a certain state which can be calculated either as (x y )z
| ih | | i
or ( y z )x . This is one aspect of the strength and versatility of the Dirac
h | i | i
notation.
5.2.3 Projectors
An important class of operators are the projectors. These are operators that
project a state onto a subspace of the Hilbert space. Suppose we have an
n-dimensional Hilbert space with an orthonormal basis i n and let (k)
{| i}i=1 { }
denoteak-dimensionalsubsetofthe basisvectors. Thenconsidertheoperators
P = i i. (5.38)
(k)
| ih |
X
(k)
Taking (k) = i n this is just the identity operator I.
{ } {| i}i=1
106
Next consider an arbitrary state x = n α j and let P act on this
| i j=1 j | i (k)
state P
n n
P x = i i α j = α i ij = α i
(k) j j i
| i | ih | | i | ih | i | i
X (k) (cid:0)Xj=1 (cid:1) X (k) Xj=1 X (k)
effectively restricting the summation to the subset (k). Thus P projects the
(k)
state onto the substate spanned by the subset of basis vectors (k).
An important property of projectionoperatorsis that acting twice with the
same projector have no further action on the state. This is almost trivial, as
canbe seen by actingon moretime with P onthe projectedstate α i .
(k) (k) i | i
Thus we find the general operator equation for projectors P
P P =P . (5.39)