Why do the d orbitals have these notations?

Why do the d orbitals have the following notations: $xy, yz, xz, z^2$ and $ x^2-y^2$? What do they represent in their wave-functions?

Answer

The notation is shorthand for the hydrogen atom $\ell = 2$ wavefunctions in real form, in cartesian coordinates. In other words, the Schrodinger equation is solved in spherical coordinates, in terms of spherical harmonics, which involve complex numbers; however, linear combinations of degenerate (equal energy) solutions are also solutions. This permits the wavefunction solutions to be expressed using real numbers only. These real solutions can be transformed to cartesian coordinates.

$z^2$ represents that the wavefunction is proportional to $(3z^2-r^2)/r^2$.

$xy$ represents that the wavefunction is proportional to $xy/r^2$.

$yz$ represents that the wavefunction is proportional to $yz/r^2$.

$xz$ represents that the wavefunction is proportional to $xz/r^2$.

$x^2 - y^2$ represents that the wavefunction is proportional to $(x^2 - y^2)/r^2$.

See this Table of Real Spherical Harmonic Functions for the a list of functions in both spherical coordinates and cartesian coordinates, for s through f orbitals.