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Chapter 4: Problem 26

Consider a system of two electrons that can have either paired or unpaired spins (e.g. a biradical). The energy of the system depends on the relative orientation of their spins. Show that the operator \(\left(h J / \hbar^{2}\right) s_{1} \cdot s_{2}\) distinguishes between singlet and triplet states. The system is now exposed to a magnetic field in the \(z\) -direction. Because the two electrons are in different environments, they experience different local fields and their interaction energy can be written \(\left(\mu_{\mathrm{B}} / \hbar\right) \times\) \(B\left(g_{1} s_{1 z}+g_{2} s_{2 z}\right)\) with \(g_{1} \neq g_{2} ; \mu_{5}\) is the Bohr magneton and \(g\) is the electron \(g\) -value, quantities discussed in Chapter 13 Establish the matrix of the total hamiltonian, and demonstrate that when \(h J \gg \mu_{\mathrm{B}} \mathscr{B},\) the coupled representation is "bctter', but that when \(\mu_{B} D\) : wh \(J\), the uncoupled representation is 'better', Find the eigenvalues of the system in each case. Hint. Use the vector coupling coefficients in Resource section 2 to determine hamiltonian matrix elements.

Short Answer

Expert verified
The operator \(hJ / \hbar^{2}\) \(s_{1} \cdot s_{2}\) does distinguish between singlet and triplet states. The Hamiltonian matrix and the eigenvalues will depend on the conditions (\(hJ\) vs. \(\mu_{B} B\)) as mentioned in the problem.

Step by step solution

01

Expressing the given operator

The operator \(hJ / \hbar^{2}\) \(s_{1} \cdot s_{2}\) can be written in terms of spin operators \(S\) as follows: \(S^{2}\) = \(\hbar^{2}\) \(S(S + 1)\) where \(S = s_{1} + s_{2}\), which can take the values of 0 (singlet state) and 1 (triplet state).
02

Calculate for Singlet and Triplet States

For the singlet state where S=0, the operator value is 0. For the triplet state where S=1, the operator value is \(2\hbar^{2}\)J. The operator, therefore, clearly distinguishes between singlet and triplet states.
03

Establish the Hamiltonian matrix

The interaction energy expressed by \(\mu_{B} / \hbar\) \(B(g_{1} s_{1z}+g_{2} s_{2z})\) indicates the Hamiltonian for the system. Write this as a 2x2 Hamiltonian matrix.
04

Finding Eigenvalues for 'Better' Coupled and Uncoupled Representations

The 'better' representation is determined by comparing \(hJ\) and \(\mu_{B} B\). For \(hJ >> \mu_{B} B\), the coupled representation applies. For \(\mu_{B} B >> hJ\), the uncoupled representation applies. Start by determining the Hamiltonian for the representation (coupled and uncoupled). Then find its eigenvalues using linear algebra techniques.

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Most popular questions from this chapter

Calculate the matrix elements (a) \(\left\langle 0,0\left|l_{z}\right| 0,0\right\rangle\) (b) \(\left\langle 2,1\left|l_{+}\right| 2,0\right\rangle\) (c) \(\left\langle 2,2\left|l_{+}^{2}\right| 2,0\right\rangle,\) (d) \(\left\langle 2,0\left|l_{+} l_{-}\right| 2,0\right\rangle\) (e) \(\left\langle 2,0\left|l \downarrow_{+}\right| 2,0\right\rangle,\) and (f) \(\left\langle 2,0\left|l^{2} l_{z} l^{2}+\right| 2,0\right\rangle\)

Calculate the values of the following matrix elements between p-orbitals: (a) \(\left\langle\mathrm{p}_{x}\left|l_{z}\right| \mathrm{p}_{y}\right\rangle,\) (b) \(\left\langle\mathrm{p}_{x}\left|l_{+}\right| \mathrm{p}_{y}\right\rangle\) \((\mathrm{c})\left\langle\mathrm{p}_{z}\left|l_{y}\right| \mathrm{p}_{x}\right\rangle\) \((d)\left\langle p_{z}\left|l_{x}\right| p_{y}\right\rangle,\) and (e) \(\left\langle\mathrm{p}_{z}\left|l_{x}\right| \mathrm{p}_{x}\right\rangle .\) Hint. Use the relations between \(\mathrm{p}_{x}, \mathrm{p}_{y}, \mathrm{p}_{z}\) and \(\mathrm{p}_{0}, \mathrm{p}_{+1}, \mathrm{p}_{-1}\).

(a) Demonstrate that if \(\left[j_{1 q}, j_{2 q^{\prime}}\right]=0\) for all \(q, q^{\prime},\) then (b) Go on to show that if \(j_{1} \times j_{1}=i \hbar j_{1}\) and \(j_{1} \times j_{2}=-j_{2} \times j_{1}\) \(j_{2} \times j_{2}=i \hbar j_{2},\) then \(j \times j=\) ihj where \(j=j_{1}+j_{2}\)

Using the Pauli matrix representation, reduce each of the operators (a) \(s_{x} s_{y}\) (b) \(s_{x} s_{y}^{2} s_{z}^{2},\) and (c) \(s_{x}^{2} s_{y}^{2} s_{z}^{2},\) to a single spin operator.

In some cases \(m_{11}\) and \(m_{12}\) may be specified at the same time as \(j\) because although \(\left[f^{2}, j_{1 z}\right]\) is non-zero, the effect of \(\left[j^{2}, j_{12}\right]\) on the state with \(m_{i 1}=j_{1}, m_{i 2}=j_{2}\) is zero. Confirm that \(\left[j^{2}, j_{1 z}\right]\left|j_{1} j_{1} ; j_{2} / j_{2}\right\rangle=0\) and \(\left[j^{2}, j_{1 z}\right]\left|j_{1},-j_{1} ; j_{2},-j_{2}\right\rangle=0\).
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