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We have seen that a functional is a machine which inputs a ket vector and
spits out a complex number. Consider a
somewhat different machine which inputs a ket
vector and spits out another ket vector in a deterministic fashion. Mathematicians
call such a machine an operator. We are only interested in operators
which preserve the linear dependencies of the ket vectors upon which
they act. Such operators are termed linear operators. Consider an operator
labeled . Suppose that when this operator acts on a general ket vector
it spits out a new ket vector which is denoted
. Operator is linear provided that

(25) 
for all ket vectors and , and

(26) 
for all complex numbers . Operators and are said to be equal
if

(27) 
for all kets in the ket space in question. Operator is termed the
null operator
if

(28) 
for all ket vectors in the space. Operators can be added together. Such addition
is defined to obey a commutative and associate algebra:
Operators can also be multiplied. The multiplication is associative:
However, in general, it is noncommutative:

(33) 
So far, we have only considered linear operators acting on ket vectors. We can also
give a meaning to their operating on bra vectors. Consider the inner product
of a general bra with the ket . This product is a
number which 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

(34) 
for any and .
The triple product of , , and
can be written
without ambiguity, provided we adopt the
convention that the bra vector always goes on the left, the operator in the middle,
and the ket vector on the right.
Consider the dual bra to . This bra depends antilinearly on
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,

(35) 
It is readily demonstrated that

(36) 
plus

(37) 
It is also easily seen that the adjoint of the adjoint of a linear operator
is equivalent to the original operator. A Hermitian operator has
the special property that it is its own adjoint: i.e.,

(38) 
Next: The outer product
Up: Fundamental concepts
Previous: Bra space
Richard Fitzpatrick
20060216