This product clearly depends linearly on the bra and the ket . Suppose that we right-multiply the previous product by the general ket . We obtain

(1.43) |

because 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 (1.42) by a general bra . It is also readily shown that

(1.44) |

(See Exercise 4.) Mathematicians term the operator the