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Chapter 12: Problem 3

Evaluate the polarizability \(\alpha_{z z}\) and polarizability volume of a hydrogen atom; for simplicity, confine the perturbation sum to the 2 p-orbitals. Use the following results for matrix elements: \\[ \begin{array}{l} \left\langle n^{\prime}, l+1, m_{l}|z| n l m_{l}\right\rangle=\left\\{\frac{(l+1)^{2}-m_{l}^{2}}{(2 l+3)(2 l+1)}\right\\}^{1 / 2}\left\langle n^{\prime}, l+1|r| n l\right\rangle \\ \langle n p|r| 1 \mathrm{s}\rangle=\left\\{\frac{2^{8} n^{7}(n-1)^{2 n-5}}{(n+1)^{2 n+5}}\right\\}^{1 / 2} a_{0} \end{array} \\]

Short Answer

Expert verified
The complete solution requires a detailed calculation using the provided matrix elements and formulas. The polarizability \(\alpha_{z z}\) and polarizability volume are properties of a hydrogen atom expressed in terms of these matrix elements. These results are then applied to derive the formula for the polarizability \(\alpha_{z z}\) and its volume for a hydrogen atom.

Step by step solution

01

Analyze the matrix elements

To start off, analyze the provided matrix elements. The first matrix element determines the wave function overlap in the spatial coordinate z between the states \(\left|n' l+1, m_{l}\right\rangle\) and \(\left|n l, m_{l}\right\rangle\). The second matrix element considers the overlap between an n p-orbital and the 1s orbital of a hydrogen atom in terms of their radial distance \(r\). Particularly look at the term \(a_{0}\), which is the Bohr radius, the average distance between the electron and the nucleus in a simple hydrogen atom.
02

Compute \(\alpha_{z z}\)

The polarizability \(\alpha_{z z}\) measures how much the electron cloud deforms in the z-direction under applied electric field \(E_{z}\). The formula for estimating polarizability is given by: \[\alpha_{z z} = 2 \left( \frac{2 \mu E_{z}}{ħ^2} \right) \sum_{n' \neq n} \frac{\left|\langle n',l+1| \hat{O} |n,l\rangle\right|^2}{E_{n} - E_{n'}}\] The factor out front is called the 'oscillator strength' describing the efficiency of a transition driven by a perturbation. The sum involves an energy denominator, which is the energy difference between the new state and the unperturbed state. Here, the operator \(\hat{O}\) and states would be those present in the provided first matrix element expression.
03

Evaluate it for hydrogen

The values of \(\left \langle n', l+1|z|n, l \right \rangle\) and \(\langle n p|r| 1 \mathrm{s}\rangle\) are given and it must be plugged into the formula above to get \(\alpha_{z z}\) for a hydrogen atom. Be sure to use the correct orbital relations, i.e. \(l = 1\), \(m_{l} = 0\) and use a perturbation sum mostly over the \(2p\) orbitals.
04

Evaluate polarizability volume

The polarizability volume measures how much an atom can be polarized by an external electric field. It can be calculated as: \[Volume = \frac{4}{3}\pi (\alpha_{z z})^{3/2}\] where \(\alpha_{z z}\) is the polarizability computed in previous steps.

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Most popular questions from this chapter

Prove the sum rule \\[ \Sigma_{p} x_{m f} x_{f m} \omega_{f n}=\left(\hbar / 2 m_{e}\right) \delta_{m n}+\frac{1}{2} \omega_{m n}\left(x^{2}\right)_{m n^{+}} \\] Hint. Consider the matrix elements of the commutator \\[ \left[H, x^{2}\right] \\].

Show group theoretically that in a tetrahedral molecule (a) the mean hyperpolarizability is zero, (b) the only non-zero components are \(\beta_{x y z}\) and the permutations of its indices. Hint. The mean is defined as \(\frac{3}{3}\left(\beta_{x x z}+\beta_{y z z}+\beta_{z z z}\right)\) and so (b) implies (a). For (b) consider the symmetry characteristics of \(E=-(1 / 3 !) \Sigma_{a, b, c} \beta_{a b c} E_{a} E_{b} E_{c}\) the generalization of eqn 12.11.

Evaluate the rotational strength of a transition of an electron from a \(2 \mathrm{p}_{x}\) -orbital to a \(2 \mathrm{p}_{z}, 3 \mathrm{d}_{x y}\) -hybrid orbital. Assume the orbitals are on a carbon atom. Estimate the optical rotation angle for 590 nm light. Hint. Follow Example \(12.4,\) with changes of detail. For carbon, take \(\zeta_{p}=1.57 / a_{0}\) and \(\zeta_{d}=0.33 / a_{0}\) and use \(\lambda_{k 0}=200 \mathrm{nm}\).
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