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Chapter 4: Problem 21

Determine what total angular momenta may arise in the following composite systems: (a) \(j_{1}=3, j_{2}=4 ;\) (b) the orbital momenta of two electrons (i) both in p-orbitals, (ii) both in d-orbitals, (iii) the configuration \(\mathrm{p}^{1} \mathrm{d}^{1} ;\) (c) the spin angular momenta of four electrons. Hint. Use the ClebschGordan series, eqn \(4.42 ;\) apply it successively in (c).

Short Answer

Expert verified
Part (a): Total angular momenta can be 1, 2, 3, 4, 5, 6, and 7. Part (b): For two electrons both in p-orbitals, total angular momentum ranges from 0 to 2; both in d-orbitals, ranges from 0 to 4; for \(\mathrm{p}^{1}\ \mathrm{d}^{1}\) configuration, ranges from 1 to 3. Part (c): Total spin angular momenta can be 0, 1 and 2.

Step by step solution

01

Solve for Part (a)

When two states with angular momenta \(j_{1}\) and \(j_{2}\) are combined, the total angular momentum is given by the quantum number \(j\), which can be represented as \(j_{1} + j_{2}\), \(j_{1} + j_{2} - 1\), ..., \(\left | j_{1} - j_{2} \right |\). As \(j_{1}=3\) and \(j_{2}=4\), total angular momentum can range from \(\left | 3 - 4 \right | = 1\) to \(3 + 4 = 7\), which includes 1, 2, 3, 4, 5, 6, and 7.
02

Solve for Part (b)

The quantum number for p-orbitals is 1 and for d-orbitals is 2. \(i) For two electrons both in p-orbitals, the total angular momentum can range from 0 to 2. \(ii) For two electrons both in d-orbitals, the total angular momentum can range from 0 to 4. \(iii) For the configuration \(\mathrm{p}^{1}\ \mathrm{d}^{1}\), the total angular momentum can range from 1 to 3.
03

Solve for Part (c)

For four electrons, consider two at a time. The spin quantum number \(s\) for an electron is 1/2. Hence for two electrons, the total spin angular momentum can be 0 or 1. Considering each pair of electrons in sequence, the total spin angular momenta can be 0, 1 and 2. Apply Clebsch-Gordan series successively to find these values.

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Most popular questions from this chapter

What are the possible outcomes of a single measurement of the \(z\) -component of spin angular momentum of an electron in the spin state \((1 / 4)^{1 / 2} \alpha-(3 / 4)^{1 / 2} \beta ?\)

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) \(\left\langle 2,0\left|l^{2} l_{z} l^{2}+\right| 2,0\right\rangle\)

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.

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 \(j=j_{1}+j_{2}\)
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