This clearly depends linearly on the bra and the ket . Suppose that we right-multiply 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 left-multiplying the expression (39) by a general bra . It is also easily demonstrated that

(41) |

Mathematicians term the operator the