Question

Upon what definitions do we base the claim that the ${\mathit{\Psi }}_{\mathbf{2}\mathbf{p}\mathbf{x}}$ and ${\mathit{\Psi }}_{\mathbf{2}\mathbf{p}\mathbf{y}}$ states of equations $\left(10-1\right)$ are related to x and y just as ${\mathit{\Psi }}_{\mathbf{2}\mathbf{p}\mathbf{z}}$ is to $\mathit{z}$.

Short answer

It is proven by converting Cartesian to spherical polar coordinates.

Step-by-step solution

A concept of polar coordinate

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.

Understanding the Similarities and Differences between these Wave Functions

States ${\Psi }_{2px}$ and ${\Psi }_{2py}$ are of the same shapes as ${\Psi }_{2pz}$ 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. $\left(\gamma ,\theta ,\varphi \right)$.

Transformation of Coordinates

The transformation between Cartesian and spherical polar coordinates is

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

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

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

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