The position operator, denoted as "x̂," operates on a wave function in quantum mechanics to determine the position of a particle. When the position operator acts on a wave function, it returns a number or a coordinate. Specifically, it returns the eigenvalue associated with the position of the particle.
In one dimension, the position operator returns a scalar value representing the position along the x-axis. In three dimensions, the position operator returns a set of three coordinates (x, y, z) representing the position in Cartesian coordinates.
It's important to note that the position operator is a mathematical operator, and when applied to a wave function, it yields a value that corresponds to the position of the particle in space.