Question #43406

An electron in a hydrogen atom is prepared in a quantum state with orbital angular momentum quantum number $l=1$, magnetic quantum number $m_l=1$, and spin magnetic quantum number $m_s=1/2$ (quantized along the $z$-axis). The atom is placed at rest at the coordinates $(x_0, 0, z_0)$ in a static, non-uniform magnetic field given by: $\vec{B} = B_0 \left( -\frac{\alpha}{2} x \hat{i} - \frac{\alpha}{2} y \hat{j} + (1 + \alpha z)\hat{k} \right)$ where $B_0$ and $\alpha$ are positive constants. Assume the atom's magnetic dipole moment vector $\vec{\mu}$ is fixed and equal to its expectation value $\langle \vec{\mu} \rangle$ along the $z$-axis. What are the magnitudes of the net magnetic force $F$ and the magnetic torque $\tau$ experienced by the atom about its center of mass? (Let $\mu_B = \frac{e\hbar}{2m_e}$ be the Bohr magneton, and assume the Landé g-factor for the electron spin is exactly 2).