Hydrogen wave functions

A.3 Hydrogen wave functions

The solutions of the Schrödinger equation for the Coulomb potential $V (r) = - ℏ c α ∕ r$ have energy $E_{n} = - \frac{1}{n^{2}} E_{Ry}$ , where $E_{Ry} = \frac{1}{2} α^{2} m c^{2} = 13.6$ eV (with $m$ the reduced mass of the electron-proton system). (Recall $α = e^{2} ∕ (4 π ε_{0} ℏ c) \approx 1 ∕ 137$ .) The spatial wavefunctions are $ψ_{n l m} (r) = R_{n, l} (r) Y_{l}^{m} (θ, ϕ)$ .

The radial wavefunctions are as follows, where $a_{0} = ℏ c ∕ (m c^{2} α)$ :

\begin{array}{rcl} R_{1, 0} (r) & = & \frac{2}{a_{0}^{3 ∕ 2}} exp (- \frac{r}{a_{0}}), \\ R_{2, 0} (r) & = & \frac{2}{{(2 a_{0})}^{3 ∕ 2}} (1 - \frac{r}{2 a_{0}}) exp (- \frac{r}{2 a_{0}}), \\ R_{2, 1} (r) & = & \frac{1}{\sqrt{3} {(2 a_{0})}^{3 ∕ 2}} \frac{r}{a_{0}} exp (- \frac{r}{2 a_{0}}), \\ R_{3, 0} (r) & = & \frac{2}{{(3 a_{0})}^{3 ∕ 2}} (1 - \frac{2 r}{3 a_{0}} + \frac{2 r^{2}}{27 a_{0}^{2}}) exp (- \frac{r}{3 a_{0}}), \\ R_{3, 1} (r) & = & \frac{4 \sqrt{2}}{9 {(3 a_{0})}^{3 ∕ 2}} \frac{r}{a_{0}} (1 - \frac{r}{6 a_{0}}) exp (- \frac{r}{3 a_{0}}), \\ R_{3, 2} (r) & = & \frac{2 \sqrt{2}}{27 \sqrt{5} {(3 a_{0})}^{3 ∕ 2}} {(\frac{r}{a_{0}})}^{2} exp (- \frac{r}{3 a_{0}}) . \end{array}

They are normalised, so $\int_{0}^{\infty} {(R_{n, l} (r))}^{2} r^{2} d r = 1$ . Radial wavefuntions of the same $l$ but diﬀerent $n$ are orthogonal (the spherical harmonics take care of orthogonality for diﬀerent $l$ s).

The following radial integrals can be proved:

\begin{array}{rcl} ⟨ r^{2} ⟩ & = & \frac{a_{0}^{2} n^{2}}{2} (5 n^{2} + 1 - 3 l (l + 1)), \\ ⟨ r ⟩ & = & \frac{a_{0}}{2} (3 n^{2} - l (l + 1)), \\ ⟨\frac{1}{r}⟩ & = & \frac{1}{n^{2} a_{0}}, \\ ⟨\frac{1}{r^{2}}⟩ & = & \frac{1}{(l + 1 ∕ 2) n^{3} a_{0}^{2}}, \\ ⟨\frac{1}{r^{3}}⟩ & = & \frac{1}{l (l + 1 ∕ 2) (l + 1) n^{3} a_{0}^{3}} . \end{array}

For hydrogen-like atoms (single-electron ions with nuclear charge $|e| Z$ ) the results are obtained by substituting $α \to Z α$ (and so $a_{0} \to a_{0} ∕ Z$ ).

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