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2.P: Exercises

  1. Demonstrate that

     

    $\displaystyle [q_i, q_j]_{cl}$ $\displaystyle = 0,$    
    $\displaystyle [p_i, p_j]_{cl}$ $\displaystyle = 0,$    
    $\displaystyle [q_i, p_j]_{cl}$ $\displaystyle = \delta_{ij},$    
     

     

    where $ [\cdots,\cdots]_{cl}$ represents a classical Poisson bracket. Here, the $ q_i$ and $ p_i$ are the coordinates and corresponding canonical momenta of a classical, many degree of freedom, dynamical system.
  2. Verify that

     

    $\displaystyle [u, v]$ $\displaystyle = - [v, u],$    
    $\displaystyle [u, c]$ $\displaystyle = 0,$    
    $\displaystyle [u_1+ u_2, v]$ $\displaystyle = [u_1, v] + [u_2, v],$    
    $\displaystyle [u, v_1 + v_2]$ $\displaystyle = [u, v_1] + [u, v_2],$    
    $\displaystyle [u_1\, u_2, v]$ $\displaystyle = [u_1, v] \,u_2 + u_1\, [u_2, v],$    
    $\displaystyle [u, v_1 \,v_2]$ $\displaystyle = [u, v_1] \,v_2 + v_1 \,[u, v_2], [u, [v, w] ]+ [v, [w, u] ] + [w, [u, v]] = 0,\nonumber$    
     

     

    where $ [\cdots,\cdots]$ represents either a classical or a quantum mechanical Poisson bracket. Here, $ u$ , $ u$ , $ w$ , etc., represent dynamical variables (i.e., functions of the coordinates and canonical momenta of a dynamical system), and $ c$ represents a number.

     

  3. Consider a Gaussian wavepacket whose corresponding wavefunction is

     

    $\displaystyle \psi(x') =\psi_0\,\exp\left[-\frac{(x'-x_0)^{\,2}}{4\,\sigma^2}\right],
$

     

    where $ \psi_0$ , $ x_0$ , and $ \sigma$ are constants. Demonstrate that Here, $ x$ and $ p$ are a position operator and its conjugate momentum operator, respectively.

     

    1.  

      $\displaystyle \langle x\rangle = x_0,$

       

    2.  

      $\displaystyle \langle ({\mit\Delta x})^2\rangle = \sigma^2,$

       

    3.  

      $\displaystyle \langle p\rangle = 0,$

       

    4.  

      $\displaystyle \langle ({\mit\Delta} p)^2\rangle = \frac{\hbar^2}{4\,\sigma^2}.
$

       

  4. Suppose that we displace a one-dimensional quantum mechanical system a distance $ a$ along the $ x$ -axis. The corresponding displacement operator is

     

    $\displaystyle D(a) = \exp\left(-{\rm i}\,p_x\,a/\hbar\right),
$

     

    where $ p_x$ is the momentum conjugate to the position operator $ x$ . Demonstrate that

     

    $\displaystyle D(a)\,x\,D(a)^{\,\dag } = x - a.
$

     

    [Hint: Use the momentum representation, $ x = {\rm i}\,\hbar\,d/dp_x$ .] Similarly, demonstrate that

     

    $\displaystyle D(a)\,x^m\,D(a)^{\,\dag } = (x-a)^m.
$

     

    Hence, deduce that

     

    $\displaystyle D(a)\,V(x)\,D(a)^{\,\dag } = V(x-a),
$

     

    where $ V(x)$ is a general function of $ x$ .

    Let $ k=p_x/\hbar$ , and let $ \vert k'\rangle$ denote an eigenket of the $ k$ operator belonging to the eigenvalue $ k'$ . Demonstrate that

     

    $\displaystyle \vert A\rangle = \sum_{n=-\infty,\infty}c_n\,\vert k'+n\,k_a\rangle,
$

     

    where the $ c_n$ are arbitrary complex coefficients, and $ k_a=2\pi/a$ , is an eigenket of the $ D(a)$ operator belonging to the eigenvalue $ \exp(-{\rm i}\,k'\,a)$ . Show that the corresponding wavefunction can be written

     

    $\displaystyle \psi_A(x') = {\rm e}^{\,{\rm i}\,k'\,x'}\,u(x'),
$

     

    where $ u(x'+a)=u(x')$ for all $ x'$ .

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