Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 1

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.

Short Answer

Expert verified
The selection rules, which can be derived from the evaluation of the electric-dipole transition moment group theoretically for one-electron atoms are as follows: \(\Delta l = \pm 1\), \(\Delta m_{l} = 0, \pm 1\), and with \(\Delta n\) having no restrictions, thus being unlimited.

Step by step solution

01

Understand the task

The task is to demonstrate the selection rules for a one-electron atom. This means proving that \(\Delta l = \pm 1\), \(\Delta m_{l} = 0, \pm 1\), and \(\Delta n\) is unlimited by evaluating the electric-dipole transition moment \(\left\langle n^{\prime}\left|m_{l}^{\prime}\right| \mu | n l m_{l}\right\rangle\). The dipole operator \(\mu\) is given by \(\mu_{x}=-e r \sin \theta \cos \varphi\), \(\mu_{y}=-e r \sin \theta \sin \varphi\), and \(\mu_{z}=-e r \cos \theta\).
02

Relate the coordinates with the components of the dipole operator

Firstly, based on the spherical grid description, the components can be written as \(Y_{lm}\) functions. The components \(\mu_{x}\), \(\mu_{y}\), and \(\mu_{z}\) are proportional to \(Y_{l m,}\) with \(l=1\). So the dipole operator can be represented by spherical harmonics.
03

Evaluation of the integral with group theory

Using the group representation, it's clear that the transition \(\left\langle n^{\prime}\left|m_{l}^{\prime}\right| \mu | n l m_{l}\right\rangle\) is nonzero only if the states \(\left| n l m_{l}\right\rangle\) and \(\left| n^{\prime} l^{\prime} m_{l}^{\prime}\right\rangle\) are in the same representation. Considering the properties of spherical harmonics, one notes that only certain values of \(\Delta l\), \(\Delta m_{l}\), and \(\Delta n\) produce nonzero results.
04

Identify selection rules from the analysis

From the analysis in Step 3, you can make the following conclusions: because the dipole operator \(\mu\) has \(l=1\), only (initial and final) states with \(\Delta l = \pm 1\) can be connected through the dipole operator. Therefore, \(\Delta l = \pm 1\). Likewise, \(\Delta m_{l}\) is limited to \(0, \pm 1\), as the dipole operator does not change the state. The only unspecified parameter is \(n\), which means that \(\Delta n\) is unlimited.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(\mathrm{On}\) 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 \(r^{2} \rho\).

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 \(V^{2}\).

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.

(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 $$\begin{array}{l} \sum_{s=0}^{n} s=\frac{1}{2} n(n+1) \quad \sum_{s=0}^{n} s^{2}=\frac{1}{6} n(n+1)(2 n+1) \\ \sum_{s=0}^{n} s^{3}=\frac{1}{4} n^{2}(n+1)^{2} \end{array}$$

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.
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Reactions

Read Explanation

Inorganic Chemistry

Read Explanation

Chemistry Branches

Read Explanation

Making Measurements

Read Explanation

Kinetics

Read Explanation

Chemical Analysis

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free