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Basic Properties of Operators
Most of the properties of operators are obvious, but they are summarized
below for completeness.
 The sum and difference of two operators
and
are given by
 The product of two operators is defined by

(35) 
 Two operators are equal if

(36) 
for all functions f.
 The identity operator
does nothing (or multiplies by 1)

(37) 
A common mathematical trick is to write this operator as a sum over a
complete set of states (more on this later).

(38) 
 The associative law holds for operators

(39) 
 The commutative law does not generally hold for operators.
In general,
.
It is convenient to
define the quantity

(40) 
which is called the commutator of
and .
Note
that the order matters, so that
.
If
and
happen to commute, then
.
 The nth power of an operator
is defined as nsuccessive applications of the operator, e.g.

(41) 
 The exponential of an operator
is defined via
the power series

(42) 
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