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So far we have formed the following products:
, ,
, ,
. Are there any other products
we are allowed to form? How about

(39) 
This clearly depends linearly on the ket and the bra .
Suppose that we rightmultiply the above product by the general ket
.
We obtain

(40) 
since
is just a number. Thus,
acting
on a general ket yields another ket. Clearly, the product
is a linear operator. This operator also acts on bras,
as is easily demonstrated by leftmultiplying the expression (39) by a general
bra . It is also easily demonstrated that

(41) 
Mathematicians term the operator
the outer product
of and . The outer product should not be confused with
the inner product,
, which is just a number.
Next: Eigenvalues and eigenvectors
Up: Fundamental concepts
Previous: Operators
Richard Fitzpatrick
20060216