Question

Verify for the angular solutions $\odot \left(\theta \right)\phi \left(\varphi \right)$of Table 7.3 that replacing $\varphi$ with $\varphi +\mathrm{\pi }$ and replacing $\theta$ with $\pi -\theta$gives the same function whenis even and the negative of the function when lis odd.

Short answer

And for all the cases when l=0 and l=2 , either both or neither of them will change signs and hence the function will remain unchanged.

Step-by-step solution

Replacing ϕ with ϕ+π :

A function which acts as a mathematical description of a quantum state of an isolated quantum system, is called a wave function.

In Azimuthal wave function,$\phi \left(\varphi \right)=e^{i{m}_{\mathcal{l}}\varphi }$,

Where, $\varphi$is the colatitude and $m_1$is the magnetic quantum number.

By replacing $\varphi$with$(\varphi +\pi )$ , you get,
role="math" localid="1659699420914" $$\begin{gathered} \phi (\varphi +\pi )=e^{i{m}_{\mathcal{l}}(\varphi +\pi )} \\ =e^{i{m}_{\mathcal{l}}\left(\pi \right)} \\ =e^{i{m}_{\mathcal{l}}\varphi }(\cos m_l\pi +i\sin m_l\pi ) \end{gathered}$$

From the above equation, you get, sine term is zero, while cosine term is + 1 while is even and cosine term is -1 when it is odd.

Hence,$\phi \left(\varphi \right)$ the changes sign when is odd and remains changed otherwise.

Replacing θ with π-θ :

In the function $\odot \left(\theta \right)$, here, $\theta$is the colatitude

By replacing $\theta$with$\pi -\theta$ you get,
$$\begin{gathered} \cos (\pi -\theta )=-\cos \left(\theta \right) \\ \sin (\pi -\theta )=\sin \left(\theta \right) \end{gathered}$$

Hence, only the terms having odd power of $\cos \left(\theta \right)$will change.