So far, we have formed the following products: , , , , . Are there any other products we are allowed to form? How about
This clearly depends linearly on the bra and the ket . Suppose that we right-multiply the above product by the general ket . We obtain
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 left-multiplying the expression (39) by a general bra . It is also easily demonstrated that
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.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)