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8.P: Exercises

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    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


      $\displaystyle \left\langle \vert\mbox{\boldmath$\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


      $\displaystyle 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


      $\displaystyle \frac{{\mit\Delta}\lambda}{\lambda} \simeq 4\times 10^{-8}.


      The only non-zero $ 1s\leftrightarrow 2p$ electric dipole matrix elements take the values


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


      where $ a_0$ is the Bohr radius.