Chapter 11, Problem 11.90E

Interpretation Introduction

Interpretation:

The forms of the five real $3d$ wavefunctions are to be stated.

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 $\cos \varphi +i\sin \varphi$.

Answer to Problem 11.90E

The forms of the five real $3d$ wavefunctions are given below.

$\begin{array}{rll}{\Psi }_{32\left(-1\right)}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \sin \varphi \\ {\Psi }_{321}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \cos \varphi \\ {\Psi }_{32\left(-2\right)}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \sin 2\varphi \\ {\Psi }_{322}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \cos 2\varphi \end{array}$

${\Psi }_{320}=\frac{1}{486}{\left(\frac{6Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\left(3{\cos }^2\theta -1\right)$

Explanation of Solution

From Table 11.4, the equations used to determine five $3d$ wavefunctions are given below.

$\begin{array}{rll}{\Psi }_{320}& =& \frac{1}{486}{\left(\frac{6Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\left(3{\cos }^2\theta -1\right)\\ {\Psi }_{32\left(-1\right)}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta e^{-i\varphi }\\ {\Psi }_{321}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta e^{i\varphi }\\ {\Psi }_{32\left(-2\right)}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta e^{-2i\varphi }\end{array}$

${\Psi }_{322}=\frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta e^{2i\varphi }$

Use Euler’s identity in the above wavefunctions as given below.

$\begin{array}{rll}{\Psi }_{32\left(-1\right)}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \left(\cos \varphi -i\sin \varphi \right)\\ {\Psi }_{321}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \left(\cos \varphi +i\sin \varphi \right)\\ {\Psi }_{32\left(-2\right)}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \left(\cos 2\varphi -i\sin 2\varphi \right)\\ {\Psi }_{322}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \left(\cos 2\varphi +i\sin 2\varphi \right)\end{array}$

The imaginary part of wavefunctions with $m_{-l}$ and the real part of wavefunctions with $m_l$ are taken to get the real forms of wavefunctions. Thus, the forms of the five real $3d$ wavefunctions can be written as follows.

$\begin{array}{rll}{\Psi }_{32\left(-1\right)}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \sin \varphi \\ {\Psi }_{321}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \cos \varphi \\ {\Psi }_{32\left(-2\right)}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \sin 2\varphi \\ {\Psi }_{322}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \cos 2\varphi \end{array}$

${\Psi }_{320}=\frac{1}{486}{\left(\frac{6Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\left(3{\cos }^2\theta -1\right)$

Conclusion

The forms of the five real $3d$ wavefunctions are given below.

$\begin{array}{rll}{\Psi }_{32\left(-1\right)}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \sin \varphi \\ {\Psi }_{321}& =& \frac{1}{81}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\sin \theta \cos \theta \cos \varphi \\ {\Psi }_{32\left(-2\right)}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \sin 2\varphi \\ {\Psi }_{322}& =& \frac{1}{162}{\left(\frac{Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}{\sin }^2\theta \cos 2\varphi \end{array}$

${\Psi }_{320}=\frac{1}{486}{\left(\frac{6Z^3}{\pi a^3}\right)}^{1/2}\frac{Z^2{r}^2}{a^2}{e}^{-Zr/3a}\left(3{\cos }^2\theta -1\right)$