Yes, if the wavefunctions are eigenfunctions of the position operator ;)

There is a position operator. The operator is just the position variable. The eigenfunction equation is

XΨ = x₀Ψ

where X is the position operator and x₀ is the eigenvalue. In configuration space, this becomes

xΨ = x₀Ψ

where x is the position variable. The eigenfunction is the Dirac delta function

Ψ(x) = δ(x - x₀).

This is a weird function. It has two key properties. First, it is zero everywhere except at the single point x=x₀, where it is infinite. Second, its integral from minus infinity to plus infinity is equal to one,

∫_-∞^∞ δ(x - x₀) dx = 1.

See http://en.wikipedia.org/wiki/Dirac_delta_function