Displacement is similar to distance, but unlike distance displacement is a vector. The displacement between two points can be expressed using the equation
$$\vec{r}=\vec{a_1}-\vec{a_2}$$ with $\vec{r}$ being the displacement between the two vectors $\vec{a_1}$, and $\vec{a_2}$. So the norm of the displacement vector is the distance.
Let's say the position of object 1 is given as $(x_1,y_1)$ while the position of object 2 is given as $(x_2,y_2)$, and so the displacement of object 1 relative to object 2 is given as $(x_1-x_2,y_1-y_2)$.
Let's also say that the probability amplitude of object 1 as a function of position is given as $f(x_1,y_1)$, and the probability amplitude of object 2 as a function of position is given as $g(x_2,y_2)$. Also the probability amplitude of the displacement of object 1 relative to object 2 is given as $h(x_1-x_2,y_1-y_2)$.
Given $f(x_1,y_1)$, and $g(x_2,y_2)$, how do I find $h(x_1-x_2,y_1-y_2)$?
The straightforward way to do it is to use the convolution: $$ h(a,b) = \iint f(x,y)g(a-x,b-y)\,dx\,dy $$