Application of the axioms—examples

Suppose we are in a state ip). Then determine the action of the operator qiy that is, 

That is, in the coordinate representation the action of g is simply a multiplication of the coordinate representation of the wave function with g.  

what is the action of g We write the operator as q, multiply from the left by the bra (g, and insert two unit operators, one between the operators and one between the operator and V ) to introduce the matrix elements of the operator in the coordinate representation that are known from the axioms. We get 

In order to use the axioms, we need to recast Eq. (F.19) such that the coordinate representation of the momentum operator appears. This is done by first multiplying 

The term [5(qi - g ) VK Ol-oo always equals zero since tp(q ) = 0 for q[ oo. The momentum operator in the coordinate representation is therefore a differential operator and the result of its action is the derivative of the wave function at qt multiplied by -ih, a standard result given in any textbook on quantum mechanics, here deduced directly from the axioms of quantum mechanics. 

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