91Ó°ÊÓ

A polar coordinate is one of two numbers that identify a point in a plane based on its distance from a fixed point on a line and the angle that line makes with the fixed line.

States ${\mathrm{Ψ}}_{2px}$ and ${\mathrm{Ψ}}_{2py}$ are of the same shapes as ${\mathrm{Ψ}}_{2pz}$ and they are all equivalent. The only difference between these states is the orientation of coordinates. These wave functions are defined based on spherical polar coordinates. $\left(\mathrm{γ},\mathrm{θ},\mathrm{ϕ}\right)$.

The transformation between Cartesian and spherical polar coordinates is

$\begin{array}{l}x=r\sin \mathrm{θ}\cos \mathrm{θ}\\ y=r\sin \mathrm{θ}\cos \mathrm{θ}\\ z=r\cos \mathrm{θ}\end{array}$

The wave functions of the 2px, 2py and 2pz states are given as

$\begin{array}{l}\mathrm{Ψ}\left(x\right)={\mathrm{Ψ}}_{2px}a\sin \mathrm{θ}\cos \mathrm{θ}\\ \mathrm{Ψ}\left(y\right)={\mathrm{Ψ}}_{2py}a\sin \mathrm{θ}\cos \mathrm{θ}\\ \mathrm{Ψ}\left(z\right)={\mathrm{Ψ}}_{2pz}a\cos \mathrm{θ}\end{array}$

Hence, from this its prove that the states 2px, 2py and 2pz depends on angles $\mathrm{θ}$and $\mathrm{ϕ}$ which are the same as that described in Cartesian coordinates for X, Y, and Z .