We have pointed out that a fundamental assertion of quantum mechanics is that we create operators of observables by requiring that they obey the commutation relations arising from the Uncertainty Principle. There are several ways in which this requirement can be met. One choice would be to let the operator for position x be simply the operator for multiplication be x. Once that is chosen, then, to satsify the commutation relation, we must assign the momentum operator as follows:
This is called the position representation. From here our mathematical task is to find the expression for any and all observables and write out their equations in classical physics in terms of x and px. This can be done. Then we substitute for these two variables the operator expressions given above. THIS IS THEN THE QUANTUM OPERATOR FOR THE OBSERVABLE.
There is another way of meeting these requirements. We can first assume that the operator for momentum is multiplication of p. Then we must have
This is called the momentum representation. Making this substitution produces another operator for the same observable property. This creates a whole new way to find eigenfunctions and eigenvalues. Though the functions and operators are different, the results are the same - if they weren't our theory wouldn't be worthwhile. Chemists usually use the position representation. Solid state physics often employs the momentum representation. There are other representations possible. We will generally focus our attention from here on out with the position representation.
As an example of how this process can form other operators, consider kinetic energy. Classically we know that
Since we have written it in a form which uses only x or p, we substitute our operators for x and p (only p in this case) and obtain
In three dimensions, we obtain the following which the "clever student' will quickly recognize as the kinetic energy part of the Hamiltonian operator.
(Remember the "del" operator?). All other operators are formed in a similar manner. Some require a bit of effort to cast in the correct form, but it can be done!