(1.26) |

for all kets and , and

for all complex numbers . Operators and are said to be equal if

(1.28) |

for all kets in the ket space in question. Operator is termed the

(1.29) |

for all kets in the space. This operator is usually denoted 0 . It follows from Equation (1.5) that

(1.30) |

where is a general operator. Operator is termed the

(1.31) |

for all kets in the space. This operator is usually denoted . Operators can be added together. Such addition is defined to obey a commutative and associate algebra: that is,

(1.32) | ||

(1.33) |

Operators can also be multiplied. Operator multiplication is associative: that is,

(1.34) | ||

(1.35) |

However, in general, operator multiplication is non-commutative: that is,

(1.36) |

So far, we have only considered linear operators acting on kets. We can also give a meaning to their operation on bras. Consider the inner product of a general bra with the ket . This product is a number that depends linearly on . Thus, it may be considered to be the inner product of with some bra. This bra depends linearly on , so we may look on it as the result of some linear operator applied to . This operator is uniquely determined by the original operator , so we might as well call it the same operator acting on . A suitable notation to use for the resulting bra when operates on is . The equation which defines this vector is

for any and . The triple product of , , and can be written without ambiguity, provided we adopt the convention that the bra always goes on the left, the operator in the middle, and the ket on the right.

Consider the dual bra to
. This bra depends antilinearly on
(i.e., if
is multiplied by the complex number
then the corresponding bra is multiplied by
) and must therefore depend linearly on
.
Thus, it may
be regarded as the result of some linear operator applied to
.
This operator is termed the *adjoint* of
, and is denoted
. Thus,

It is readily demonstrated that

(1.39) |

plus

(1.40) |

It is also easily seen that the adjoint of the adjoint of a linear operator is equivalent to the original operator. (See Exercise 4.) An

(1.41) |

Obviously, a complex number can be regarded as a trivial operator that modifies the length and phase of a ket upon which it acts, without changing the ket's direction. Furthermore, it follows from Equations (1.17) and (1.27) that a complex number operator, , commutes with any other operator, and that its adjoint is . Finally, it is easily appreciated that the identity operator corresponds to the number unity, while the null operator corresponds to the number zero.