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 :
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.