Exercises

1. Demonstrate that ${\bf p}\cdot{\bf A}={\bf A}\cdot{\bf p}$ when $\nabla\cdot{\bf A} = 0$ , where ${\bf p}$ is the momentum operator, and ${\bf A}({\bf x})$ is a real function of the position operator, ${\bf x}$ . Hence, show that the Hamiltonian (870) is Hermitian.

2. Find the selection rules for the matrix elements $\langle n,l,m\vert\,x\,\vert n',l',m'\rangle$ , $\langle n,l,m\vert\,y\,\vert n',l',m'\rangle$ , and $\langle n,l, m\vert\,z\,\vert n',l',m'\rangle$ to be non-zero. Here, $\vert n,l,m\rangle$ denotes an energy eigenket of a hydrogen-like atom corresponding to the conventional quantum numbers, $n$ , $l$ , and $m$ .

3. Demonstrate that
   $$\left\langle \vert\mbox{\boldmath$\epsilon$}\cdot{\bf f}_{21}\vert^{\,2}\right\rangle = \frac{f_{21}^{\,2}}{3},$$
   where the average is taken over all directions of the incident radiation.

4. Demonstrate that the spontaneous decay rate (via an electric dipole transition) from any 2p state to a 1s state of a hydrogen atom is
   $$w_{2p\rightarrow 1s} = \left(\frac{2}{3}\right)^8\alpha^5\,\frac{m_e\,c^2}{\hbar}=6.26\times 10^8\,{\rm s}^{-1},$$
   where $\alpha$ is the fine structure constant. Hence, deduce that the natural line width of the associated spectral line is
   $$\frac{{\mit\Delta}\lambda}{\lambda} \simeq 4\times 10^{-8}.$$
   The only non-zero $1s\leftrightarrow 2p$ electric dipole matrix elements take the values

   $$\langle 1,0,0\vert\,x\,\vert 2,1,\pm 1\rangle = \pm\frac{2^7}{3^5}\,a_0,$$

   $$\langle 1,0,0\vert\,y\,\vert 2,1,\pm 1\rangle = {\rm i}\,\frac{2^7}{3^5}\,a_0,$$

   $$\langle 1,0,0\vert\,z\,\vert 2,1,0\rangle = \sqrt{2}\,\frac{2^7}{3^5}\,a_0,$$

   
   
   where $a_0$ is the Bohr radius.