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Operator Product

  One could now consider defining multiplication of operators in the same way, with the result of the operation of the operator to be the product of the result of the application of each of the operators separately. However, a much more common combination of two operators is to apply them separately in a sequence of two successive operations. Amazingly, this second definition preserves nearly all of the familiar algebra of multiplication! We will proceed with this definition and define the product of two operators to be the result of their sequential application,

This procedure is sensible since returns a new function which may then be fed to as input to produce the final output defined as .

Like the familiar multiplication of numbers, the function operator product is also associative, as we see by studying the action of a three-way product on an arbitrary function ,

As the operator product is associative, without ambiguity we may write the product without parenthesis. As with normal multiplication, raising an operator to an integer power is just a short-hand for repeated application of the operator,

 

Note that in our notation there is a subtle difference between and . The former is the product of n factors of as defined in (10) above, while the later is a single operator, defined physically so that the quantum average is given correctly. Although, they will turn out to be equal, the definitions of and are entirely different.

There is one key property which our operator product does not share with common multiplication. Defined as the sequential application of two operations, there is no reason to expect that the operator product be commutative, that the net affect of two operations be independent of the sequence in which they are applied. Symbolically, we expect in general that . This property of operators is closely related to the Heisenberg uncertainty principle. The drive to give this basic principle of quantum physics a direct expression in our mathematical formalism is a key force in our choice of the definition of operator product in terms of sequential application. Because of the prime importance of the non-commutivity of the operator product, we will dedicate the entire next section to the issue of commutivity.

Before discussing non-commutivity, there are two final properties of the operator product which mimic normal multiplication and with which the student should be familiar. First, as with common multiplication, there is an operator multiplicative identity. It is just the constant operator discussed in the previous subsection. It is easily verified, using the procedures above that . (Note that one must consider both a left and a right identity as the operator product is not commutative.) One may also show that multiplication by yields the additive inverse of an operator, (.

The final familiar property of the operator product is the distributive property,

 

The ``commuted'' form of this statement, , does not follow directly. While true for quantum operators, it is not true for general operators and depends on a property known as linearity which we shall discuss in detail in section (4.1).



next up previous
Next: Commutators Up: Combining Simple Operators: Previous: Operator Sum



Prof. Tomas Alberto Arias
Wed Oct 11 21:37:35 EDT 1995