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:

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