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 $i=1,2,3$ , 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, (3.55), 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 $E$ : that is,
      $$H\,\vert E\rangle = E\,\vert E\rangle.$$
      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 $n=0,1,2,\cdots$ . Here, $A$ and $A^\dag$ are termed ladder operators. To be more exact, $A$ is termed a lowering operator (because it lowers the energy quantum number, $n$ , by unity), whereas $A^\dag$ is termed a raising operator (because it raises the energy quantum number by unity).

   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

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

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

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

      $$\phantom{=}\left.+ (2\,n+1)\,\delta_{n'\,n}\right].$$

      
      

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

      $$\phantom{=}\left.+ (2\,n+1)\,\delta_{n'\,n}\right].$$

      
      

   5. 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 the one-dimensional quantum harmonic oscillator discussed in the previous two exercises. Let $\vert\alpha\rangle$ be a properly normalized eigenket of the lowering operator, $A$ , corresponding to the eigenvalue $\alpha$ , where $\alpha$ can be any complex number. The corresponding state is known as a coherent state.
   1. Demonstrate that

      $$\langle x\rangle = \sqrt{\frac{2\,\hbar}{m\,\omega}}\,{\rm Re}(\alpha),$$

      $$\langle p_x\rangle = \sqrt{2\,m\,\hbar\,\omega}\,{\rm Im}(\alpha),$$

      $$\langle x^{\,2}\rangle = \frac{2\,\hbar}{m\,\omega}\left(\frac{1}{4}+\left[{\rm Re}(\alpha)\right]^2\right),$$

      $$\langle p_x^{\,2}\rangle = 2\,m\,\hbar\,\omega\left(\frac{1}{4}+\left[{\rm Im}(\alpha)\right]^2\right),$$

      
      
      where the expectation values are relative to the coherent state. Hence, deduce that
      $$\langle ({\mit\Delta}x)^2\rangle\, \langle ({\mit\Delta p_x})^2\rangle =\frac{\hbar^{\,2}}{4}.$$
      In other words, a coherent state is characterized by the minimum possible uncertainty in position and momentum.

   2. If $\vert n\rangle$ is the properly normalized energy eigenket belonging to the energy eigenvalue $E_n= (n+1/2)\,\hbar\,\omega$ then show that
      $$\vert\alpha\rangle=\sum_{n=0,\infty} c_n\,\vert n\rangle,$$
      where
      $$c_n = \frac{\alpha^{\,n}}{\sqrt{n!}}\,\exp\left(-\frac{\vert\alpha\vert^{\,2}}{2}\right).$$

   3. Show that the expectation value of the energy for the coherent state is
      $$\langle H\rangle = (\vert\alpha\vert^{\,2}+1/2)\,\hbar\,\omega.$$

   4. Putting in time dependence, so that
      $$\vert n,t\rangle ={\rm e}^{-{\,\rm i}\,E_n\,t/\hbar}\,\vert n\rangle,$$
      where $\vert n\rangle\equiv \vert n,0\rangle$ , demonstrate that $\vert\alpha,t\rangle$ remains an eigenket of $A$ , but that the eigenvalue evolves in time as
      $$\alpha(t)= {\rm e}^{-{\rm i}\,\omega\,t}\,\alpha.$$
      Hence, deduce that
      $$\vert\alpha,t\rangle= {\rm e}^{-{\rm i}\,\omega\,t/2}\,\vert\alpha\rangle.$$

   5. Writing
      $$\alpha = \sqrt{\frac{m\,\omega}{2\,\hbar}}\,a,$$
      where $a$ is real and positive, show that

      $$\langle x\rangle = a\,\cos(\omega\,t),$$

      $$\langle p_x\rangle = -m\,\omega\,a\,\sin(\omega\,t).$$

      
      
      Of course, these expressions are analogous to those of a classical harmonic oscillator of amplitude $a$ and angular frequency $\omega$ . This suggests that a coherent state of a quantum harmonic oscillator is the state that most closely imitates the behavior of a classical oscillator.

   6. Show that the properly normalized wavefunction corresponding to the state $\vert\alpha,t\rangle$ takes the form

      $$\psi(x',t)=\psi_0\left(x'-\sqrt{\frac{2\,\hbar}{m\,\omega}}\,\alpha(t)\right),$$ (3.111)

      
      
      where $\psi_0(x')$ is the properly normalized, stationary, ground-state wavefunction.

6. 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.

7. 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
   $$[D_x(a),H] = 0,$$
   where $D_x(a)$ is the displacement operator defined in Exercise 6. 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 Bloch's theorem.

8. 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.

9. Consider a particle in one dimension whose Hamiltonian is
   $$H = \frac{p_x^{\,2}}{2\,m} + V(x).$$
   Suppose that the particle is in a 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
   $$\left\langle \frac{p_x^{\,2}}{2\,m}\right\rangle = -E + \langle V\rangle+\left\langle x\,\frac{dV}{dx}\right\rangle.$$
   Here, $E$ is the energy eigenvalue. [Hint: You may assume, without loss of generality, that the stationary wavefunction is real.] Hence, prove the Virial theorem,
   $$\left\langle \frac{p_x^{\,2}}{2\,m}\right\rangle =\frac{1}{2}\left\langle x\,\frac{dV}{dx}\right\rangle.$$

10. Consider a particle of mass $m$ and charge $q$ moving in the $x$ -$y$ plane in the presence of the uniform perpendicular magnetic field ${\bf B} = B_z\,{\bf e}_z$ . Demonstrate that the Hamiltonian of the system can be written
    $$H = \hbar\,\omega\left(\frac{1}{2}+ A^\dag\,A\right),$$
    where $\omega = q\,B_z/m$ , and
    $$A = \frac{{\mit\Pi}_x+{\rm i}\,{\mit\Pi}_y}{\sqrt{2\,\hbar\,q\,B_z}}.$$
    In addition, show that
    $$[A, A^\dag ] = 1.$$
    Hence, deduce that the possible energy eigenstates of the particle are
    $$E_n = (n+1/2)\,\hbar\,\omega,$$
    where $n$ is a non-negative integer. These energy levels are known as Landau levels.

11. Show that the time-dependent Schrödinger equation
    $${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t}=\frac{1}{2\,m}\,(-{\rm i}\,\hbar\,\nabla' -q\,{\bf A})^2\,\psi + q\,\phi\,\psi,$$
    where $\psi=\psi({\bf x}',t)$ , ${\bf A}= {\bf A}({\bf x}',t)$ , and $\phi=\phi({\bf x}',t)$ , can be written
    $${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t}=\frac{1}{2\,m}\,(... ... {\rm i}\,\hbar\,q\,\nabla'\cdot{\bf A} + q^{\,2}\,A^2)\,\psi + q\,\phi\,\psi.$$
    Hence, deduce that if the so-called Coloumb gauge [49],
    $$\nabla'\cdot{\bf A} = 0,$$
    is adopted then the equation simplifies to
    $${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^{\... ...{\bf A}\cdot\nabla'\psi + \frac{q^{\,2}}{2\,m}\,A^{\,2}\,\psi + q\,\phi\,\psi.$$
    Demonstrate that this equation is associated with a probability conservation law of the form
    $$\frac{\partial\rho}{\partial t} +\nabla' \cdot{\bf j} = 0,$$
    where
    $$\rho({\bf x}',t)= \vert\psi({\bf x}',t)\vert^{\,2},$$
    and
    $${\bf j}({\bf x}', t) = -\left(\frac{{\rm i}\,\hbar}{2\,m}\right)[... ...\nabla'\psi)^\ast\,\psi]- \frac{q}{m}\,\rho({\bf x}', t)\,{\bf A}({\bf x}',t).$$
    Finally, show that $\rho$ and ${\bf j}$ are invariant under a gauge transformation.