Chapter 11, Problem 11.89E

Interpretation Introduction

Interpretation:

The given equation for the complete Schrӧdinger equation for a hydrogen-like atom is to be verified.

Concept introduction:

The wavefunction for $\text{3-D}$ rotational motion involves spherical polar coordinates to explain the motion in three dimensions. It is given by the expression as follows.

${\Psi }_{m,l}=\frac{1}{\sqrt{2\pi }}{e}^{im\varphi }{\theta }_{l,m_l}$

Where, ${\Psi }_m$ represents the wavefunction dependent on quantum number $m$ and $l$. The values of $m$ and $l$ start from $0$. Also, they can take up both negative and positive integral values.According to Euler’s identity, $e^{im\varphi }$ can be written as $\cos \varphi +i\sin \varphi$.