(Solved): At time \( t=0 \), the normalized wave function for the hydrogen atom is \[ \Psi(\vec{r}, 0)=\frac ...

At time \( t=0 \), the normalized wave function for the hydrogen atom is \[ \Psi(\vec{r}, 0)=\frac{1}{\sqrt{10}}\left(2 \Psi_{100}+\Psi_{210}+\sqrt{2} \Psi_{211}+\sqrt{3} \Psi_{21,-1}\right) \] where \( \Psi_{n ! m} \) denote the normalized energy eigenfunctions of the Coulomb potential with principal quantum number \( n \) and angular momentum quantum number \( l \) and \( m \). Find: (a) the expectation value of the energy of the system; (b) the probability of the system with \( l=1 \) and \( m=-1 \); (c) write down \( \Psi(\vec{r}, t) \) at some later time \( t \). Note that the energy eigenvalue is \( E_{n}=-13.6 \mathrm{eV} / n^{2} \) since we are concerned with a hydrogen atom.