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Chapter 6: Problem 20

A hydrogen atom in a \(2 \mathrm{s}^{1}\) configuration passes into a region where it experiences an electric field in the \(z\) -direction for a time \(\tau .\) What is its electric dipole moment during its exposure and after it emerges? Hint. Use eqn 6.62 with \(\omega_{21}=0 ;\) the dipole moment is the expectation value of \(-e z\) \\[\text { use } \int \psi_{2 \mathrm{s}} z \psi_{2 \mathrm{p}} \mathrm{d} \tau=3 a_{0}\\]

Short Answer

Expert verified
The electric dipole moment is \(- 3e a_0\) during its exposure and zero when it emerges from the electric field.

Step by step solution

01

Define the Electric Dipole Moment

The electric dipole moment of a hydrogen atom is the expectation value of the product of the charge (\(e\)) and the position (\(z\)). In quantum mechanics, expectation values are computed as follows: \(\langle O \rangle = \int \psi^* O \psi \, d\tau\), where \(O\) is the operator corresponding to the observable (in this case, \(-e z\)), and \(\psi\) is the wave function.
02

Compute the Expectation Value

For this problem, we have two wave functions: \(\psi_{2s}\) before the atom interacts with the electric field, and \(\psi_{2p}\) after. The problem has specifically given \(\int \psi_{2s} z \psi_{2p} \, d\tau = 3 a_0\). Thus, as per formula, expectation value \(\langle -e z \rangle = -e \int \psi_{2 s}^* z \psi_{2 p} \, d\tau = -e (3 a_0)\) where \(a_0\) is the atomic scale factor (Bohr's radius).
03

Result and Interpretation

Therefore, the electric dipole moment during exposure to the electric field is given by \(\langle -e z \rangle = - 3e a_0\). After the atom emerges from the electric field, it will return to its original state, giving a dipole moment of zero, as there is no net separation of charges in the hydrogen atom in its natural state.

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Most popular questions from this chapter

Calculate the second-order energy correction to the ground state of a particle in a one-dimensional box for a perturbation of the form \(H^{(1)}=-\varepsilon \sin (\pi x / L)\) by using the closure approximation. Infer a value of \(\Delta E\) by comparison with the numerical calculation in Example \(6.4 .\) These two problems \((6.14 \text { and } 6.15)\) show that the parameter \(\Delta E\) depends on the perturbation and is not simply a characteristic of the system itself.

Repeat the last problem but set \(H_{\mathrm{s}, \mathrm{s}}=\gamma\) and \(S_{\mathrm{s}^{\prime}} \neq 0\) Evaluate the overlap integrals between 1 s-orbitals on centres separated by \(R ;\) use $$S=\left\\{1+\frac{R}{a_{0}}+\frac{1}{3}\left(\frac{R}{a_{0}}\right)^{2}\right\\} \mathrm{e}^{-R / a_{0}}$$ Suppose that \(\beta / \gamma=S_{\mathrm{s}, \mathrm{s}_{2}} / S_{\mathrm{s}, \mathrm{s}} .\) For a numerical result, take \(R=80 \mathrm{pm}, a_{0}=53 \mathrm{pm}\)

A simple calculation of the energy of the helium atom supposes that each electron occupies the same hydrogenic 1 s-orbital (but with \(Z=2\) ). The electronelectron interaction is regarded as a perturbation, and calculation gives $$\int \psi_{1 s}^{2}\left(r_{1}\right)\left(\frac{e^{2}}{4 \pi \varepsilon_{0} r_{12}}\right) \psi_{1 s}^{2}\left(r_{2}\right) \mathrm{d} \tau=\frac{5}{4}\left(\frac{e^{2}}{4 \pi \varepsilon_{0} a_{0}}\right)$$ (see Example 7.2 ). Estimate (a) the binding energy of helium, (b) its first ionization energy. Hint. Use eqn 6.15 with \(E_{1}=E_{2}=E_{1 \mathrm{s}} .\) Be careful not to count the electronelectron interaction energy twice.

\(\mathrm{A}\) biradical is prepared with its two electrons in a singlet state. A magnetic field is present, and because the two electrons are in different environments their interaction with the field is \(\left(\mu_{\mathrm{B}} / \hbar\right) \mathscr{B}\left(g_{1} s_{1 z}+g_{2} s_{2 z}\right)\) with \(g_{1} \neq g_{2} .\) Evaluate the time-dependence of the probability that the electron spins will acquire a triplet configuration (that is, the probability that the \(S=1, M_{s}=0\) state will be populated). Examine the role of the energy separation \(h J\) of the singlet state and the \(M_{S}=0\) state of the triplet. Suppose \(g_{1}-g_{2}\) \(\approx 1 \times 10^{-3}\) and \(J \approx 0 ;\) how long does it take for the triplet state to emerge when \(\mathscr{B}=1.0 \mathrm{T}\) ? Hint. Use eqn 6.63 ; take \(|0,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta-\beta \alpha)\) and \(|1,0\rangle=\left(1 / 2^{1 / 2}\right)(\alpha \beta+\beta \alpha) .\) See Problem 4.26 for the significance of \(\mu_{\mathrm{B}}\) and \(g\)

Suppose that the potential energy of a particle on a ring depends on the angle \(\varphi\) as \(H^{(1)}=\varepsilon \sin ^{2} \varphi .\) Calculate the first- order corrections to the energy of the degenerate \(m_{l}=\pm 1\) states, and find the correct linear combinations for the perturbation calculation. Find the second-order correction to the energy. Hint. This is an example of degenerate-state perturbation theory, and so find the correct linear combinations by solving eqn 6.42 after deducing the energies from the roots of the secular determinant. For the matrix elements, express \(\sin \varphi\) as \((1 / 2 \mathrm{i})\left(\mathrm{e}^{\mathrm{i} \varphi}-\mathrm{e}^{-\mathrm{i} \varphi}\right)\) When evaluating eqn \(6.42,\) do not forget the \(m_{1}=0\) state lying beneath the degenerate pair. The energies are equal to \(m_{l}^{2} \hbar^{2} / 2 m r^{2} ;\) use \(\psi_{m_{l}}=(1 / 2 \pi)^{1 / 2} \mathrm{e}^{i m_{\mu} \varphi}\) for the unperturbed states.
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