Exercises

1. Demonstrate that

   $$[q_i, q_j]_{cl} = 0,$$

   $$[p_i, p_j]_{cl} = 0,$$

   $$[q_i, p_j]_{cl} = \delta_{ij},$$

   
   
   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

   $$[u, v] = - [v, u],$$

   $$[u, c] = 0,$$

   $$[u_1+ u_2, v] = [u_1, v] + [u_2, v],$$

   $$[u, v_1 + v_2] = [u, v_1] + [u, v_2],$$

   $$[u_1\, u_2, v] = [u_1, v] \,u_2 + u_1\, [u_2, v],$$

   $$[u, v_1 \,v_2] = [u, v_1] \,v_2 + v_1 \,[u, v_2], [u, [v, w] ]+ [v, [w, u] ] + [w, [u, v]] = 0,\nonumber$$

   
   
   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
   $$\psi(x') =\psi_0\,\exp\left[-\frac{(x'-x_0)^{\,2}}{4\,\sigma^2}\right],$$
   where $\psi_0$ , $x_0$ , and $\sigma$ are constants. Demonstrate that

   1. $$\langle x\rangle = x_0,$$

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

   3. $$\langle p\rangle = 0,$$

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

   Here, $x$ and $p$ are a position operator and its conjugate momentum operator, respectively.

4. Suppose that we displace a one-dimensional quantum mechanical system a distance $a$ along the $x$ -axis. The corresponding displacement operator is
   $$D(a) = \exp\left(-{\rm i}\,p_x\,a/\hbar\right),$$
   where $p_x$ is the momentum conjugate to the position operator $x$ . Demonstrate that
   $$D(a)\,x\,D(a)^{\,\dag } = x - a.$$
   [Hint: Use the momentum representation, $x = {\rm i}\,\hbar\,d/dp_x$ .] Similarly, demonstrate that
   $$D(a)\,x^m\,D(a)^{\,\dag } = (x-a)^m.$$
   Hence, deduce that
   $$D(a)\,V(x)\,D(a)^{\,\dag } = V(x-a),$$
   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

   $$\vert A\rangle = \sum_{n=-\infty,\infty}c_n\,\vert k'+n\,k_a\rangle,$$
   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
   $$\psi_A(x') = {\rm e}^{\,{\rm i}\,k'\,x'}\,u(x'),$$
   where $u(x'+a)=u(x')$ for all $x'$ .