Exercises

1. Let ${\bf x}\equiv (x_1, x_2, x_3)$ be a set of Cartesian position operators, and let ${\bf p}\equiv (p_1, p_2, p_3)$ be the corresponding momentum operators. Demonstrate that

   $$[x_i, F({\bf x},{\bf p})] = {\rm i}\,\hbar \,\frac{\partial F}{\partial p_i},$$

   $$[p_i, G({\bf x},{\bf p})] = - {\rm i}\,\hbar\,\frac{\partial G}{\partial x_i},$$

   
   
   where $i=1,3$ , and $F({\bf x},{\bf p})$ , $G({\bf x},{\bf p})$ are functions that can be expanded as power series.

2. Assuming that the potential $V({\bf x})$ is complex, demonstrate that the Schrödinger time-dependent wave equation, (274), can be transformed to give
   $$\frac{\partial \rho}{\partial t} + \nabla'\cdot {\bf j} = 2\,\frac{{\rm Im}(V)}{\hbar}\,\rho,$$
   where
   $$\rho({\bf x}',t)= \vert\psi({\bf x}',t)\vert^{\,2},$$
   and
   $${\bf j}({\bf x}', t) = \left(\frac{\hbar}{m} \right){\rm Im} (\psi^\ast \,\nabla' \psi).$$

3. Consider one-dimensional quantum harmonic oscillator whose Hamiltonian is
   $$H = \frac{p_x^{\,2}}{2\,m}+ \frac{1}{2}\,m\,\omega^2\,x^2,$$
   where $x$ and $p_x$ are conjugate position and momentum operators, respectively, and $m$ , $\omega$ are positive constants.
   1. Demonstrate that the expectation value of $H$ , for a general state, is positive definite.

   2. Let
      $$A = \sqrt{\frac{m\,\omega}{2\,\hbar}}\,x + {\rm i}\,\frac{p_x}{\sqrt{2\,m\,\omega\,\hbar}}.$$
      Deduce that

      $$[A,A^\dag ] = 1,$$

      $$H =\hbar\,\omega\left(\frac{1}{2} + A^\dag\,A\right),$$

      $$[H,A] =-\hbar\,\omega\,A,$$

      $$[H,A^\dag ] = \hbar\,\omega\,A^\dag .$$

      
      

   3. Suppose that $\vert E\rangle$ is an eigenket of the Hamiltonian whose corresponding energy is $E$ : i.e.,
      $$H\,\vert E\rangle = E\,\vert E\rangle.$$
      Demonstrate that

      $$H\,A\,\vert E\rangle = (E-\hbar\,\omega)\,A\,\vert E\rangle,$$

      $$H\,A^\dag\,\vert E\rangle = (E+\hbar\,\omega)\,A^\dag\,\vert E\rangle.$$

      
      
      Hence, deduce that the allowed values of $E$ are
      $$E_n = (n+1/2)\,\hbar\,\omega,$$
      where $n=0,1,2,\cdots$ .

   4. Let $\vert E_n\rangle$ be a properly normalized (i.e., $\langle E_n\vert E_n\rangle = 1$ ) energy eigenket corresponding to the eigenvalue $E_n$ . Show that the kets can be defined such that

      $$A\,\vert E_n\rangle = \sqrt{n}\,\vert E_{n-1}\rangle,$$

      $$A^\dag\,\vert E_n\rangle = \sqrt{n+1}\,\vert E_{n+1}\rangle.$$

      
      
      Hence, deduce that
      $$\vert E_n\rangle = \frac{1}{\sqrt{n!}}\,(A^\dag )^n\,\vert E_0\rangle.$$

   5. Let the $\psi_n(x')=\langle x'\vert E_n\rangle$ be the wavefunctions of the properly normalized energy eigenkets. Given that
      $$A\,\vert E_0\rangle = \vert\rangle,$$
      deduce that
      $$\left(\frac{x'}{x_0}+x_0\,\frac{d}{d x'}\right)\psi_0(x')= 0,$$
      where $x_0=(\hbar/m\,\omega)^{1/2}$ . Hence, show that
      $$\psi_n(x') = \frac{1}{\pi^{1/4}\,(2^n\,n!)^{1/2}\,x_0^{\,n+1/2}}\... ...c{d}{dx'}\right)^n\exp\left[-\frac{1}{2}\left( \frac{x'}{x_0}\right)^2\right].$$

4. Consider the one-dimensional quantum harmonic oscillator discussed in Exercise 3. Let $\vert n\rangle$ be a properly normalized energy eigenket belonging to the eigenvalue $E_n$ . Show that

   $$\langle n'\vert\,x\,\vert n\rangle = \sqrt{\frac{\hbar}{2\,m\,\omega}}\left(\sqrt{n}\,\delta_{n'\,n-1}+\sqrt{n+1}\,\delta_{n'\,n+1}\right),$$

   $$\langle n'\vert\,p_x\,\vert n\rangle = {\rm i}\sqrt{\frac{m\,\hbar\,\omega}{2}}\left(-\sqrt{n}\,\delta_{n'\,n-1}+\sqrt{n+1}\,\delta_{n'\,n+1}\right),$$

   $$\langle n'\vert\,x^2\,\vert n\rangle = \left(\frac{\hbar}{2\,m\,\omega}\right)\left(\sqrt{n\,(n-1)}\,\... ...'\,n-2}+\sqrt{(n+1)\,(n+2)}\,\delta_{n'\,n+2}+ (2\,n+1)\,\delta_{n'\,n}\right),$$

   $$\langle n'\vert\,p_x^{\,2}\,\vert n\rangle = \left(\frac{m\,\hbar\,\omega}{2}\right)\left(-\sqrt{n\,(n-1)}\,... ...n'\,n-2}-\sqrt{(n+1)\,(n+2)}\,\delta_{n'\,n+2}+ (2\,n+1)\,\delta_{n',n}\right).$$

   
   
   Hence, deduce that
   $$\langle({\mit\Delta} x)^2\rangle\,\langle ({\mit\Delta}p_x)^2\rangle = (n+1/2)^2\,\hbar^2$$
   for the $n$ th eigenstate.

5. Consider a particle in one dimension whose Hamiltonian is
   $$H = \frac{p_x^{\,2}}{2\,m} + V(x).$$
   By calculating $[[H,x], x]$ , demonstrate that
   $$\sum_{n'} \vert\langle n\vert\,x\,\vert n'\rangle\vert^{\,2}\,(E_{n'}-E_{n})= \frac{\hbar^2}{2\,m},$$
   where $\vert n\rangle$ is a properly normalized energy eigenket corresponding to the eigenvalue $E_n$ , and the sum is over all eigenkets.

6. Consider a particle in one dimension whose Hamiltonian is
   $$H = \frac{p_x^{\,2}}{2\,m} + V(x).$$
   Suppose that the potential is periodic, such that
   $$V(x-a)= V(x),$$
   for all $x$ . Deduce that
   $$[D(a),H] = 0,$$
   where $D(a)$ is the displacement operator defined in Exercise 4. Hence, show that the wavefunction of an energy eigenstate has the general form
   $$\psi(x') = {\rm e}^{\,{\rm i}\,k'\,x'}\,u(x'),$$
   where $k'$ is a real parameter, and $u(x'-a)=u(x')$ for all $x'$ . This result is known as the Bloch theorem.

7. Consider the one-dimensional quantum harmonic oscillator discussed in Exercise 3. Show that the Heisenberg equations of motion of the ladder operators $A$ and $A^\dag$ are

   $$\frac{dA}{dt} = -{\rm i}\,\omega\,A,$$

   $$\frac{dA^\dag }{dt} = {\rm i}\,\omega\,A^\dag ,$$

   
   
   respectively. Hence, deduce that the momentum and position operators evolve in time as

   $$p_x(t) = \cos(\omega\,t) \,p_x(0)- m\,\omega\,\sin(\omega\,t)\,x(0),$$

   $$x(t) = \cos(\omega\,t)\,x(0) + \frac{\sin(\omega\,t)}{m\,\omega}\,p_x(0),$$

   
   
   respectively, in the Heisenberg picture.

8. Consider a one-dimensional stationary bound state. Using the time-independent Schrödinger equation, prove that
   $$\left\langle \frac{p_x^{\,2}}{2\,m}\right\rangle = E- \langle V\rangle,$$
   and
   $$\left\langle \frac{p_x^{\,2}}{2\,m}\right\rangle = -E + \langle V\rangle+\left\langle x\,\frac{dV}{dx}\right\rangle.$$
   [Hint: You can assume, without loss of generality, that the stationary wavefunction is real.] Hence, prove the Virial theorem,
   $$\left\langle \frac{p_x^{\,2}}{2\,m}\right\rangle =\frac{1}{2}\left\langle x\,\frac{dV}{dx}\right\rangle.$$