Exercises

1. Demonstrate that

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

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

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

   Here, $[\cdots,\cdots]_{cl}$ represents a classical Poisson bracket. Moreover, 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

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

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

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

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

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

   6. $$[u, v_1 \,v_2] = [u, v_1] \,v_2 + v_1 \,[u, v_2].$$

   7. $$[u, [v, w]]+ [v, [w, u]] + [w, [u, v]] = 0.$$

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

3. Let $\xi$ be an operator whose eigenvalues $\xi'$ can take a continuous range of values. Let the $\vert\xi'\rangle$ be the corresponding eigenstates. Let $f(\xi)$ be a function of $\xi$ that can be expanded as a power series. Demonstrate that
   $$[f(\xi),\,\xi] = 0,$$
   and
   $$f(\xi)\,\vert\xi'\rangle = f(\xi')\,\vert\xi'\rangle,$$
   where $f(\xi')$ is the same function of the eigenvalue $\xi'$ that $f(\xi)$ is of the operator $\xi$ . Let $g(\eta)$ be a function of the operator $\eta$ that can be expanded as a power series, and let $\xi$ and $\eta$ commute. Demonstrate that
   $$[f(\xi), g(\eta)] =0.$$

4. 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 real numbers. Demonstrate that

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

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

   3. $$\langle p_x\rangle = 0.$$

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

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

5. Let $D_x({\mit\Delta} x)$ and $D_y({\mit\Delta} y)$ be operators that displace a quantum mechanical system the finite distances ${\mit\Delta} x$ and ${\mit\Delta}y$ along the $x$ - and $y$ -directions, respectively. Demonstrate that
   $$D_x({\mit\Delta} x_2)\,D_x({\mit\Delta} x_1)= D_x({\mit\Delta} x_1)\,D_x({\mit\Delta} x_2)= D_x({\mit\Delta}x_2+ {\mit\Delta}x_1),$$
   and
   $$D_x({\mit\Delta} x)\,D_y({\mit\Delta} y)= D_y({\mit\Delta} y)\,D_x({\mit\Delta} x).$$
   What are the physical significances of these results?

6. Suppose that we displace a one-dimensional quantum mechanical system a finite distance $a$ along the $x$ -axis. The corresponding operator is
   $$D_x(a) = \exp\left(\frac{-{\rm i}\,p_x\,a}{\hbar}\right),$$
   where $p_x$ is the momentum conjugate to the position operator $x$ . Demonstrate that
   $$D_x(a)\,x\,D_x(a)^{\,\dag } = x - a.$$
   [Hint: Use the momentum representation, $x = {\rm i}\,\hbar\,d/dp_x$ .] Similarly, demonstrate that
   $$D_x(a)\,x^{\,m}\,D_x(a)^{\,\dag } = (x-a)^{\,m},$$
   where $m$ is a non-negative integer. Hence, deduce that
   $$D_x(a)\,V(x)\,D_x(a)^{\,\dag } = V(x-a),$$
   where $V(x)$ is a function of $x$ that can be expanded as a power series.

   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 $D_x(a)$ 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 $x'$ .