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 $\hat{A}$ and $\hat{B}$ are given by
  

  $$(\hat{A} + \hat{B}) f = \hat{A} f + \hat{B} f$$ (33)

  $$(\hat{A} - \hat{B}) f = \hat{A} f - \hat{B} f$$ (34)

  
  

• The product of two operators is defined by
  

  $$\hat{A} \hat{B} f \equiv \hat{A} [ \hat{B} f ]$$ (35)

  
  

• Two operators are equal if
  

  $$\hat{A} f = \hat{B} f$$ (36)

  
  
  for all functions $f$.

• The identity operator $\hat{1}$ does nothing (or multiplies by 1)
  

  $${\hat 1} f = f$$ (37)

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

  $$\sum_i \vert i \rangle \langle i \vert f = f$$ (38)

  
  

• The associative law holds for operators
  

  $$\hat{A}(\hat{B}\hat{C}) = (\hat{A}\hat{B})\hat{C}$$ (39)

  
  

• The commutative law does not generally hold for operators. In general, $\hat{A} \hat{B} \neq \hat{B} \hat{A}$. It is convenient to define the quantity
  

  $$[\hat{A}, \hat{B}]\equiv \hat{A} \hat{B} - \hat{B} \hat{A}$$ (40)

  
  
  which is called the commutator of $\hat{A}$ and $\hat{B}$. Note that the order matters, so that $[ \hat{A}, \hat{B}] = - [ \hat{B}, \hat{A}]$. If $\hat{A}$ and $\hat{B}$ happen to commute, then $[\hat{A}, \hat{B}] = 0$.

• The n-th power of an operator $\hat{A}^n$ is defined as $n$ successive applications of the operator, e.g.
  

  $$\hat{A}^2 f = \hat{A} \hat{A} f$$ (41)

  
  

• The exponential of an operator $e^{\hat{A}}$ is defined via the power series
  

  $$e^{\hat{A}} = \hat{1} + \hat{A} + \frac{\hat{A}^2}{2!} + \frac{\hat{A}^3}{3!} + \cdots$$ (42)