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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
 = (33) = (34)

• 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 n-th 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)

Next: Linear Operators Up: Operators Previous: Operators and Quantum Mechanics