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Physics LibreTexts

3.P: 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

     

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

     

    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

     

    $\displaystyle \frac{\partial \rho}{\partial t} + \nabla'\cdot {\bf j} = 2\,\frac{{\rm Im}(V)}{\hbar}\,\rho,
$

     

    where

     

    $\displaystyle \rho({\bf x}',t)= \vert\psi({\bf x}',t)\vert^{\,2},
$

     

    and

     

    $\displaystyle {\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

     

    $\displaystyle 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

       

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

       

      Deduce that

       

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

       

       

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

       

      $\displaystyle H\,\vert E\rangle = E\,\vert E\rangle.
$

       

      Demonstrate that

       

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

       

      Hence, deduce that the allowed values of $ E$ are

       

      $\displaystyle 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

       

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

       

      Hence, deduce that

       

      $\displaystyle \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

       

      $\displaystyle A\,\vert E_0\rangle = \vert\rangle,
$

       

      deduce that

       

      $\displaystyle \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

       

      $\displaystyle \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

     

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

     

    Hence, deduce that

     

    $\displaystyle \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

     

    $\displaystyle H = \frac{p_x^{\,2}}{2\,m} + V(x).
$

     

    By calculating $ [[H,x], x]$ , demonstrate that

     

    $\displaystyle \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

     

    $\displaystyle H = \frac{p_x^{\,2}}{2\,m} + V(x).
$

     

    Suppose that the potential is periodic, such that

     

    $\displaystyle V(x-a)= V(x),
$

     

    for all $ x$ . Deduce that

     

    $\displaystyle [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

     

    $\displaystyle \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

     

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

     

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

     

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

     

    respectively, in the Heisenberg picture.

     

  8. Consider a one-dimensional stationary bound state. Using the time-independent Schrödinger equation, prove that

     

    $\displaystyle \left\langle \frac{p_x^{\,2}}{2\,m}\right\rangle = E- \langle V\rangle,
$

     

    and

     

    $\displaystyle \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,

     

    $\displaystyle \left\langle \frac{p_x^{\,2}}{2\,m}\right\rangle =\frac{1}{2}\left\langle x\,\frac{dV}{dx}\right\rangle.
$

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