The wavefunctions for the 2p_{x} and 2p_{y} atomic orbitals for the H-atom can be written as,

Notice that the 2p_{x} and 2p_{y} atomic orbitals only differ in the sign of the complex number term for the phi exponential. Below Figure shows the polar coordinates (r, ȹ, θ) where r is the radial distance from the origin at the centre of a sphere; θ (theta) varies from 0 at the “north pole” to it π radians at the “south pole”; and ȹ (phi) varies around the “equator” from 0 to 2π radians.

Rewrite the sum of the two wavefunctions for 2p_{x} and 2p_{y} atomic orbitals for the H-atom as a real number.

**Solution**

The wavefunctions for the 2p_{x} and 2p_{y} atomic orbitals for the H-atom can be written as,

To rewrite these as purely real numbers we make use of Euler’s formulae written in terms of i ȹ,

Adding our two wavefunctions,

Substituting using Euler’s formulae gives the sum of the 2p_{x} and 2p_{y} atomic orbitals for the hydrogen atom which is a real function.

In quantum chemistry, it is often more convenient to use complex-functions, particularly when one needs to apply quantum mechanical operators to a wavefunction to generate a new wavefunction.