Suggested languages for you:

Americas

Europe

2CQ

Expert-verifiedFound in: Page 413

Book edition
2nd Edition

Author(s)
Randy Harris

Pages
633 pages

ISBN
9780805303087

**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 ${\mathit{z}}$.**

It is proven by converting Cartesian to spherical polar coordinates.

**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 ${\Psi}_{2px}$ 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. $\left(\gamma ,\theta ,\varphi \right)$.

The transformation between Cartesian and spherical polar coordinates is

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

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

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

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 .

94% of StudySmarter users get better grades.

Sign up for free