Chapter 7

Demonstrate that for one-electron atoms the selection rules are \(\Delta l=\pm 1, \Delta m_{l}=0,\pm 1,\) and \(\Delta n\) unlimited. Hint. Evaluate the electric-dipole transition moment \(\left\langle n^{\prime}\left|m_{l}^{\prime}\right| \mu | n l m_{l}\right\rangle\) with \(\mu_{x}=-e r \sin \theta \cos \varphi, \mu_{y}=-e r \sin \theta \sin \varphi,\) and \(\mu_{z}=-e r \cos \theta\). The easiest way of evaluating the angular integrals is to recognize that the components just listed are proportional to \(Y_{l m,}\) with \(l=1,\) and to analyse the resulting integral group theoretically.

Suppose that an electron experiences a shielded Coulomb potential (a Coulomb potential modified by \(\left.\operatorname{afactor} \exp \left(-r / r_{\mathrm{D}}\right), \text { where } r_{\mathrm{D}} \text { is a constant }\right) .\) Evaluate the ratio of spin-orbit coupling constants \(\zeta_{2 \mathrm{p}} / \zeta_{2 \mathrm{p}}^{\circ},\) where \(\zeta_{2 \mathrm{p}}^{\circ}\) is the constant for the unshielded potential. Explore the result of setting \(r_{\mathrm{D}}=k a_{0},\) where \(k\) is a variable parameter.

(a) Calculate the energy difference between the levels with the greatest and smallest values of \(j\) for given \(l\) and \(s\) Each term of a level is \((2 j+1)\) -fold degenerate. (b) Demonstrate that the barycentre (mean energy) of a term is the same as the energy in the absence of spin-orbit coupling. Hint. Weight each level with \(2 j+1\) and sum the energies given in eqn 7.24 from \(j=|l-s|\) to \(j=l+s\) Use the relations $$\begin{array}{l} \sum_{s=0}^{n} s=\frac{1}{2} n(n+1) \quad \sum_{s=0}^{n} s^{2}=\frac{1}{6} n(n+1)(2 n+1) \\ \sum_{s=0}^{n} s^{3}=\frac{1}{4} n^{2}(n+1)^{2} \end{array}$$

Find the first-order corrections to the energies of the hydrogen atom that result from the relativistic mass increase of the electron. Hint. The energy is related to the momentum by \(E=\left(p^{2} c^{2}+m^{2} c^{4}\right)^{1 / 2}+V .\) When \(p^{2} c^{2} \ll m^{2} c^{4}\), \(E \approx \mu c^{2}+p^{2} / 2 \mu+V-p^{4} / 8 \mu^{3} c^{2},\) where the reduced mass \(\mu\) has replaced \(m\). Ignore the rest energy \(\mu c^{2},\) which simply fixes the zero. The term \(-p^{4} / 8 \mu^{3} c^{2}\) is a perturbation; hence calculate \(\left\langle n l m_{l}\left|H^{(1)}\right| n l m_{l}\right\rangle=-\left(1 / 2 \mu c^{2}\right)\left\langle n l m_{l}\left|\left(p^{2} / 2 \mu\right)^{2}\right| n l m_{l}\right\rangle\) \(=-\left(1 / 2 \mu c^{2}\right)\left\langle n l m_{l}\left|\left(E_{n l m_{l}}-V\right)^{2}\right| n l m_{l}\right\rangle .\) We know \(E_{n l m} ;\) therefore calculate the matrix elements of \(V=-e^{2} / 4 \pi \varepsilon_{0} r\) and \(V^{2}\).

Take a trial function for the helium atom as \(\psi=\) \(\psi(1) \psi(2),\) with \(\psi(1)=\left(\zeta^{3} / \pi\right)^{1 / 2} \mathrm{e}^{-\zeta r_{1}}\) and \(\psi(2)=\left(\zeta^{3} / \pi\right)^{1 / 2} \mathrm{e}^{-\zeta_{2}}, \zeta\) being a parameter, and find the best ground-state energy for a function of this form, and the corresponding value of \(\zeta\). Calculate the first and second ionization energies. Hint. Use the variation theorem. All the integrals are standard; the electron repulsion term is calculated in Example 7.2 Interpret \(Z\) in terms of a shielding constant. The experimental ionization energies are \(24.58 \mathrm{eV}\) and \(54.40 \mathrm{eV}\).

Consider a one-dimensional square well containing two electrons. One electron has \(n=1\) and the other has \(n=2 .\) Plot a two-dimensional contour diagram of the probability distribution of the electrons when their spins are (a) parallel, (b) antiparallel. Devise a measure of the radius of the Fermi hole. Hint. Recall the discussion in Section \(7.11 .\) When the spins are parallel (for example, \(\alpha \alpha\) ) the antisymmetric combination \(\psi_{1}(1) \psi_{2}(2)-\psi_{2}(1) \psi_{1}(2)\) must be used, and when the spins are antiparallel, the symmetric combination must be used. In each case plot \(\psi^{2}\) against axes labelled \(x_{1}\) and \(x_{2}\). Computer graphics may be used to obtain striking diagrams, but a sketch is sufficient.

An excited state of atomic calcium has the electron configuration \(1 \mathrm{s}^{2} 2 \mathrm{s}^{2} 2 \mathrm{p}^{6} 3 \mathrm{s}^{2} 3 \mathrm{p}^{6} 3 \mathrm{d}^{1} 4 \mathrm{f}^{1}\) (a) Derive all the term symbols (with the appropriate specifications of \(S, L \text { and } J)\) for the electron configuration. (b) Which term symbol corresponds to the lowest energy of this electron configuration? (c) Consider a \(^{3} \mathrm{F}_{2}\) level of calcium derived from a different electron configuration than that shown above. Which of the term symbols determined in part (a) can participate in spectroscopic transitions to this \(^{3} \mathrm{F}_{2}\) level?

\(\mathrm{On}\) the basis of the Thomas-Fermi model of an atom, evaluate the radius within which there is a 50 per cent probability of finding the electron density and evaluate it for the Period 2 elements. Hint. Use the radial distribution function \(r^{2} \rho\).