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4.P: Exercises

  1. Last updated
    08:26, 16 Nov 2014
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  1. Demonstrate directly from the fundamental commutation relations for angular momentum, (300), that $ [L^2, L_z] = 0$ , $ [L^\pm, L_z] = \mp \,\hbar\,L^\pm$ , and $ [L^+,L^-] = 2\,\hbar\,L_z$ .

     

  2. Demonstrate from Equations (363)-(368) that

     

    $\displaystyle L_x$ $\displaystyle = {\rm i}\,\hbar\,\left(\sin\varphi\, \frac{\partial}{\partial \theta} + \cot\theta \cos\varphi\,\frac{\partial}{\partial \varphi}\right),$    
    $\displaystyle L_y$ $\displaystyle = -{\rm i} \,\hbar\,\left(\cos\varphi\, \frac{\partial}{\partial\theta} -\cot\theta \sin\varphi \,\frac{\partial}{\partial \varphi}\right),$    
    $\displaystyle L_z$ $\displaystyle = -{\rm i}\,\hbar\,\frac{\partial}{\partial\varphi},$    
     

     

    where $ \theta$ , $ \varphi$ are conventional spherical polar angles.

     

  3. A system is in the state $ \psi(\theta,\varphi)=Y_{l\,m}(\theta,\varphi)$ . Evaluate $ \langle L_x\rangle$ , $ \langle L_y\rangle$ , $ \langle L_x^{\,2}\rangle$ , and $ \langle L_y^{\,2}\rangle$ .

     

  4. Derive Equations (385) and (386) from Equation (384).

     

  5. Find the eigenvalues and eigenfunctions (in terms of the angles $ \theta$ and $ \varphi$ ) of $ L_x$ .

     

  6. Consider a beam of particles with $ l=1$ . A measurement of $ L_x$ yields the result $ \hbar$ . What values will be obtained by a subsequent measurement of $ L_z$ , and with what probabilities? Repeat the calculation for the cases in which the measurement of $ L_x$ yields the results 0 and $ -\hbar$ .

     

  7. The Hamiltonian for an axially symmetric rotator is given by

     

    $\displaystyle H = \frac{L_x^{\,2}+L_y^{\,2}}{2\,I_1} + \frac{L_z^{\,2}}{2\,I_2}. $

     

    What are the eigenvalues of $ H$ ?

     

  8. The expectation value of $ f({\bf x},{\bf p})$ in any stationary state is a constant. Calculate

     

    $\displaystyle 0= \frac{d}{dt}\,(\langle{\bf x}\cdot{\bf p}\rangle) = \frac{{\rm i}}{\hbar}\,\langle[H, {\bf x}\cdot{\bf p}]\rangle $

     

    for a Hamiltonian of the form

     

    $\displaystyle H = \frac{p^2}{2\,m} + V(r). $

     

    Hence, show that

     

    $\displaystyle \left\langle\frac{p^2}{2\,m}\right\rangle = \frac{1}{2}\left\langle r\,\frac{dV}{dr}\right\rangle $

     

    in a stationary state. This is another form of the Virial theorem. (See Exercise 8.)

     

  9. Use the Virial theorem of the previous exercise to prove that

     

    $\displaystyle \left\langle \frac{1}{r}\right\rangle = \frac{1}{n^2\,a_0} $

     

    for an energy eigenstate of the hydrogen atom.

     

  10. Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:

    1.  

      $\displaystyle R_{1\,0}(r) = \frac{2}{a_0^{\,3/2}}\,\exp\left(-\frac{r}{a_0}\right). $

       

    2.  

      $\displaystyle R_{2\,0}(r)= \frac{2}{(2\,a_0)^{3/2}}\left(1-\frac{r}{2\,a_0}\right)\exp\left(-\frac{r}{2\,a_0}\right). $

       

    3.  

      $\displaystyle R_{2\,1}(r)= \frac{1}{\sqrt{3}\,(2\,a_0)^{3/2}}\,\frac{r}{a_0}\,\exp\left(-\frac{r}{2\,a_0}\right). $

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