In algebra, or pre-calc, you learn that you can change the position of a function by modifying the argument. In quantum physics this idea is used to displace wave functions. If a function starts off at one position, and moves to another position, all that is needed is a change in argument. However, quantum physics likes to use linear operators to alter functions. What would an operator look like if it can change the argument of a function? In this post, I will construct a 1-D example of such an operator.

A general wave function can be written as: , where the shape of is dependent on the spatial variable .

To translate a function by distance , modify the argument of ,

To move the function right by ,

To move the function left by ,

Let’s just take the example, and without loss of generality say that can be positive or negative.

Next we can take advantage of Taylor expansions.

A function can be expanded around a point :

Here, in our example, we want to expand around , to express the translated function in terms of the original function :

Note: .

A more complete version of this expression would be:

This sum’s structure is similar to the Taylor expansion of the exponential function.

Every operator in the expansion can be reduced into an simplified operator:

This new operator can be expanded to return to what we had before: .

So the translated function is a version of the original function with a specific type of interference. Such that the structure is: .

We can reduce the operators in the interference terms into an exponential in the usual way:

The expectation value characterizes the average overlap between and .

The expectation value of the translation operator is then unity plus the expectation value of the interference operator.

The interference expectation value is shown to be an expectation value of a function of the momentum operator.

In the case of localized waves, as gets much greater than the width of , the interference term approaches -1, since the overlap between the original and translated wave function decreases. This is equivalent to the original and displaced wave function becoming more and more orthogonal.

limit

limit