## Step 1: Identify the Terms in the Hamiltonian The Hamiltonian for two identical interacting spin-½ particles in a magnetic field is given by: \[ \hat{H} = \left(\frac{eB}{m}\right) (\hat{S}_z^{(1)} + \hat{S}_z^{(2)}) + \frac{4\beta}{\hbar^2} (\hat{S}^{(1)} \cdot \hat{S}^{(2)}) \] **Explanation:** - The first term, \(\left(\frac{eB}{m}\right) (\hat{S}_z^{(1)} + \hat{S}_z^{(2)})\), represents the Zeeman effect, which is the energy of the spins in the external magnetic field along the z-axis. - The second term, \(\frac{4\beta}{\hbar^2} (\hat{S}^{(1)} \cdot \hat{S}^{(2)})\), represents the interaction between the two spins. --- ## Step 2: Expand the Interaction Term Expanding \(\hat{S}^{(1)} \cdot \hat{S}^{(2)}\) in terms of components, we get: \[ \hat{S}^{(1)} \cdot \hat{S}^{(2)} = \hat{S}_x^{(1)} \otimes \hat{S}_x^{(2)} + \hat{S}_y^{(1)} \otimes \hat{S}_y^{(2)} + \hat{S}_z^{(1)} \otimes \hat{S}_z^{(2)} \] So the Hamiltonian becomes: \[ \hat{H} = \left(\frac{eB}{m}\right) (\hat{S}_z^{(1)} + \hat{S}_z^{(2)}) + \frac{4\beta}{\hbar^2} \left[\hat{S}_x^{(1)} \otimes \hat{S}_x^{(2)} + \hat{S}_y^{(1)} \otimes \hat{S}_y^{(2)} + \hat{S}_z^{(1)} \otimes \hat{S}_z^{(2)}\right] \] **Explanation:** - The interaction is written as a sum of tensor products of the spin operators for each particle in the x, y, and z directions. --- ## **Final Answer** The Hamiltonian for two identical interacting spin-½ particles in a magnetic field is: \[ \boxed{ \hat{H} = \left(\frac{eB}{m}\right) (\hat{S}_z^{(1)} + \hat{S}_z^{(2)}) + \frac{4\beta}{\hbar^2} \left[\hat{S}_x^{(1)} \otimes \hat{S}_x^{(2)} + \hat{S}_y^{(1)} \otimes \hat{S}_y^{(2)} + \hat{S}_z^{(1)} \otimes \hat{S}_z^{(2)}\right] } \] This Hamiltonian includes both the Zeeman effect and the spin-spin interaction.