91Ó°ÊÓ

Use the vector model of angular momentum to derive the value of the angle between the vectors representing (a) two \(\alpha\) spins, (b) an \(\alpha\) and a \(\beta\) spin in a state with \(S=1\) and \(M_{\mathrm{S}}=+1\) and \(M_{\mathrm{S}}=0,\) respectively.

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.

What is the expectation value of the \(z\) -component of orbital angular momentum of electron 1 in the \(\left|G, M_{L}\right\rangle\) state of the configuration \(\mathrm{d}^{2}\) ? Hint. Express the coupled state in terms of the uncoupled states, find \(\left\langle\mathrm{G}, M_{L}\left|l_{1 z}\right| \mathrm{G}, M_{L}\right\rangle\) in terms of the vector coupling coefficients, and evaluate it for \(M_{L}=+4,+3, \ldots,-4\)

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