Basic Properties of Operators

• 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)