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Chapter 6: Problem 16

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.

Short Answer

Expert verified
Specific numerical correction values depend on quantities like \(\varepsilon\) which is not given in the problem. However, the steps to compute the first and second order corrections involve finding an appropriate linear combination of degenerate states that best matches the perturbation, and then using this combination and the terms defined at the beginning to calculate the corrections.

Step by step solution

01

Defining the Terms

Define the terms to be used for the computation. Here, \(\varepsilon\) is the potential defined by \(H^{(1)}=\varepsilon \sin ^{2} \varphi\). 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.
02

Linear Combination for the Perturbation Calculation

To find a correct linear combination for the perturbation calculation, one should solve Eq. \(6.42\). This calculation is crucial because of the presence of the degenerate \(m_{l}=\pm 1\) states which suggests the use of degenerate perturbation theory, in which one needs to find a linear combination of degenerate states that best matches the perturbation.
03

First Order Correction Calculation

Obtain the matrix elements of the perturbation. The hint suggests expressing \(sin \varphi\) as \((1 / 2 \mathrm{i})\left(\mathrm{e}^{\mathrm{i} \varphi}-\mathrm{e}^{-\mathrm{i} \varphi}\right)\). Make sure to include all the states (including \(m_l = 0\)) in the calculation, and then compute the first order correction to the energy.
04

Second Order Correction Calculation

Compute the second order correction to the energy. This is done by using the the states and energies from the previous steps and applying the formulas of perturbation theory. Simplify until we cannot simplify any further. This step may involve some algebra and meticoulous calculations.

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Most popular questions from this chapter

Show group-theoretically that when a perturbation of the form \(H^{(1)}=a z\) is applied to a hydrogen atom, the 1 s-orbital is contaminated by the admixture of \(n \mathrm{p}_{z^{-}}\) orbitals. Deduce which orbitals mix into (a) \(2 p_{x}\) -orbitals, (b) \(2 \mathrm{p}_{z}\) -orbitals (c) \(3 d_{x y}\) -orbitals.

Find the complete dependence of the \(A\) and \(B\) coefficients on atomic number for the \(2 \mathrm{p} \rightarrow 1\) s transitions of hydrogenic atoms. Calculate how the stimulated emission rate depends on \(Z\) when the atom is exposed to black-body radiation at \(1000 \mathrm{K} .\) Hint. The relevant density of states also depends on \(Z\).

Consider the hypothetical linear \(\mathrm{H}_{3}\) molecule. The wavefunctions may be modelled by expressing them as \(\psi=c_{\Lambda} s_{A}+c_{B} s_{B}+c_{C} s_{C}\) the \(s_{i}\) denoting hydrogen 1 s-orbitals of the relevant atom. Use the Rayleigh-Ritz method to find the optimum values of the coefficients and the energies of the orbital. Make the approximations \(H_{\mathrm{ss}}=\alpha, H_{\mathrm{ss}^{\prime}}=\beta\) for neighbours but 0 for non-neighbours, \(S_{\mathrm{ss}}=1,\) and \(S_{\mathrm{ss}^{\prime}}=0\) Hint. Although the basis can be used as it stands, it leads to a \(3 \times 3\) determinant and hence to a cubic equation for the energies. A better procedure is to set up symmetry-adapted combinations, and then to use the vanishing of \(H_{i j}\) unless \(\Gamma^{(i)}=\Gamma^{(j)}\).

\(\mathrm{A}\) 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 \(g\)

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}\\]
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