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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 .

• 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 successive 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   Contents
David Sherrill 2006-08-15