The wave functions for the three states with the dot plots shown in Fig. 39-23, which have $n=2, \ell=1$, and $m_{t}=0,+1$, and $-1$, are
$$
\begin{gathered}
\psi_{210}(r, \theta)=(1 / 4 \sqrt{2 \pi})\left(a^{-3 / 2}\right)(r / a) e^{-r / 2 a} \cos \theta \\
\psi_{21+1}(r, \theta)=(1 / 8 \sqrt{\pi})\left(a^{-3 / 2}\right)(r / a) e^{-r / 2 a}(\sin \theta) e^{+i \phi}, \\
\psi_{21-1}(r, \theta)=(1 / 8 \sqrt{\pi})\left(a^{-3 / 2}\right)(r / a) e^{-r / 2 a}(\sin \theta) e^{-i \phi},
\end{gathered}
$$
in which the subscripts on $\psi(r, \theta)$ give the values of the quantum numbers $n, \ell, m_{\ell}$ and the angles $\theta$ and $\phi$ are defined in Fig. 39-22. Note that the first wave function is real but the others, which involve the imaginary number $i$, are complex. Find the radial probability density $P(r)$ for (a) $\psi_{210}$ and (b) $\psi_{21+1}$ (same as for $\psi_{21-1}$ ). (c) Show that each $P(r)$ is consistent with the corresponding dot plot in Fig. 39-23. (d) Add the radial probability densities for $\psi_{210}$, $\psi_{21+1}$, and $\psi_{21-1}$ and then show that the sum is spherically symmetric, depending only on $r$.