Answer to Question #115487 in Physical Chemistry for Utkarsh

Question

How are spherical polar coordinates related to the rectangular cartesian coordinates? Illustrate
giving suitable relations. Also write the Schrodinger equation for hydrogen atom in spherical polar coordinates

Answer 1

Answer:

The spherical polar coordinates $r$, $\theta$ and $\phi$ are related to Cartesian coordinates $x$, $y$ and $z$ via the following equations:

$x = r\text{ sin} \theta\text{ cos} \phi$

$y = r\text{ sin} \theta\text{ sin}\phi$

$z = r \text{ cos}\theta$ .

And vice versa:

$r = \sqrt{x^2+y^2+z^2}$

$\phi = \text{arctan} \frac{y}{x}$

$\theta =\text{arctan}\frac{\sqrt{x^2+y^2}}{z}$ .

The time-independent Schrödinger equation in spherical polar coordinates for hydrogen atom is:

$\{-\frac{\hbar^2}{2\mu r^2}[\frac{∂}{∂r}(r^2\frac{∂}{∂r}) + \frac{1}{\text{sin}\theta}\frac{∂}{∂\theta}(\text{sin}\theta\frac{∂}{∂\theta}) + \frac{1}{\text{sin}^2\theta}\frac{∂^2}{∂\phi^2}] - \frac{e^2}{4\pi\epsilon_0r}\}\psi(r,\theta,\phi) = E\psi(r,\theta,\phi)$

Solving this equation involves the separation of the variables:

$\Psi(r,\theta,\phi) = R(r)P(\theta)F(\phi)$ .