Question #43623

In a semi-classical model, the electron (mass $m_e$) in the ground state of a hydrogen atom is treated as a continuous mass cloud rather than a point particle. The mass density $\rho_m(\vec{r})$ of this cloud is given by $\rho_m(\vec{r}) = m_e |\Psi(\vec{r})|^2$, where the normalized wave function is $\Psi(\vec{r}) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}$ and $a_0$ is the Bohr radius. An external impulsive torque $\vec{\tau}(t) = J_0 \delta(t) \hat{k}$ is applied to this electron cloud, where $J_0$ is a constant angular impulse and $\delta(t)$ is the Dirac delta function. Assuming the cloud rotates as a rigid body about the z-axis immediately after the impulse, what is the acquired angular velocity $\vec{\omega}$?