Show that the hydrogen wave function \psi_{211} is normalized.
Strategy We refer to Equation (6.8) in Chapter 6 where we normalized the wave function in one dimension. Now we want to normalize the wave function in three dimensions in spherical polar coordinates. The normalization condition is
\int_{-\infty}^{\infty} \Psi *(x, t) \Psi(x, t) d x=1 (6.8)
\int \psi_{n \ell m_{\ell}}^{*} \psi_{n \ell m_{\ell}} d \tau=1=\int \psi_{211}^{*} \psi_{211} r^{2} \sin \theta d r d \theta d \phi (7.18)
where d \tau=r^{2} \sin \theta d r d \theta d \phi is the volume element. We look up the wave function \psi_{211} using Tables 7.1 and 7.2.
\psi_{211}=R_{21} Y_{11}=\left[\frac{r}{a_{0}} \frac{e^{-r / 2 a_{0}}}{\sqrt{3}\left(2 a_{0}\right)^{3 / 2}}\right]\left[\frac{1}{2} \sqrt{\frac{3}{2 \pi}} \sin \theta e^{i \phi}\right]
| Table 7.1 Hydrogen Atom Radial Wave Functions | ||
| n | \ell | R_{n \ell}(r) |
| 1 | 0 | \frac{2}{\left(a_{0}\right)^{3 / 2}} e^{-r / a_{0}} |
| 2 | 0 | \left(2-\frac{r}{a_{0}}\right) \frac{e^{-r / 2 a_{0}}}{\left(2 a_{0}\right)^{3 / 2}} |
| 2 | 1 | \frac{r}{a_{0}} \frac{e^{-r / 2 a_{0}}}{\sqrt{3}\left(2 a_{0}\right)^{3 / 2}} |
| 3 | 0 | \frac{1}{\left(a_{0}\right)^{3 / 2}} \frac{2}{81 \sqrt{3}}\left(27-18 \frac{r}{a_{0}}+2 \frac{r^{2}}{a_{0}^{2}}\right) e^{-r / 3 a_{0}} |
| 3 | 1 | \frac{1}{\left(a_{0}\right)^{3 / 2}} \frac{4}{81 \sqrt{6}}\left(6-\frac{r}{a_{0}}\right) \frac{r}{a_{0}} e^{-r / 3 a_{0}} |
| 3 | 2 | \frac{1}{\left(a_{0}\right)^{3 / 2}} \frac{4}{81 \sqrt{30}} \frac{r^{2}}{a_{0}^{2}} e^{-r / 3 a_{0}} |
| Table 7.2 Normalized Spherical Harmonics Y[\theta, \phi] | ||
| \ell | m_{\ell} | Y_{\ell m_{\ell}} |
| 0 | 0 | \frac{1}{2 \sqrt{\pi}} |
| 1 | 0 | \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos \theta |
| 1 | \pm 1 | \mp \frac{1}{2} \sqrt{\frac{3}{2 \pi}} \sin \theta e^{\pm i \phi} |
| 2 | 0 | \frac{1}{4} \sqrt{\frac{5}{\pi}}\left(3 \cos ^{2} \theta-1\right) |
| 2 | \pm 1 | \mp \frac{1}{2} \sqrt{\frac{15}{2 \pi}} \sin \theta \cos \theta e^{\pm i \phi} |
| 2 | \pm 2 | \frac{1}{4} \sqrt{\frac{15}{2 \pi}} \sin ^{2} \theta e^{\pm 2 i \phi} |
| 3 | 0 | \frac{1}{4} \sqrt{\frac{7}{\pi}}\left(5 \cos ^{3} \theta-3 \cos \theta\right) |
| 3 | \pm 1 | \mp \frac{1}{8} \sqrt{\frac{21}{\pi}} \sin \theta\left(5 \cos ^{2} \theta-1\right) e^{\pm i \phi} |
| 3 | \pm 2 | \frac{1}{4} \sqrt{\frac{105}{2 \pi}} \sin ^{2} \theta \cos \theta e^{\pm 2 i \phi} |
| 3 | \pm 3 | \mp \frac{1}{8} \sqrt{\frac{35}{\pi}} \sin ^{3} \theta e^{\pm 3 i \phi} |
