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Next: Orbital Angular Momentum Up: Quantum Dynamics Previous: Flux Quantization and the

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=1,2,3$ , 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

    $\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$ : that is,

      $\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$ . 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

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

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

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

    5. 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 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

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

      where the expectation values are relative to the coherent state. Hence, deduce that

      $\displaystyle \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

      $\displaystyle \vert\alpha\rangle=\sum_{n=0,\infty} c_n\,\vert n\rangle,
$

      where

      $\displaystyle 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

      $\displaystyle \langle H\rangle = (\vert\alpha\vert^{\,2}+1/2)\,\hbar\,\omega.
$

    4. Putting in time dependence, so that

      $\displaystyle \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

      $\displaystyle \alpha(t)= {\rm e}^{-{\rm i}\,\omega\,t}\,\alpha.
$

      Hence, deduce that

      $\displaystyle \vert\alpha,t\rangle= {\rm e}^{-{\rm i}\,\omega\,t/2}\,\vert\alpha\rangle.
$

    5. Writing

      $\displaystyle \alpha = \sqrt{\frac{m\,\omega}{2\,\hbar}}\,a,
$

      where $ a$ is real and positive, show that

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

      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

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

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

  6. Consider a particle in one dimension whose Hamiltonian is

    $\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.

  7. 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_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

    $\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 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

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

    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.

  9. Consider a particle in one dimension whose Hamiltonian is

    $\displaystyle 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

    $\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.
$

    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,

    $\displaystyle \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

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

    where $ \omega = q\,B_z/m$ , and

    $\displaystyle A = \frac{{\mit\Pi}_x+{\rm i}\,{\mit\Pi}_y}{\sqrt{2\,\hbar\,q\,B_z}}.
$

    In addition, show that

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

    Hence, deduce that the possible energy eigenstates of the particle are

    $\displaystyle 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

    $\displaystyle {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t}=\frac{1}{2\,m}\,(-{\rm i}\,\hbar\,\nabla' -q\,{\bf A})^2\,\psi + q\,\phi\,\psi,$

    where $ \psi=\psi({\bf x}',t)$ , $ {\bf A}= {\bf A}({\bf x}',t)$ , and $ \phi=\phi({\bf x}',t)$ , can be written

    $\displaystyle {\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],

    $\displaystyle \nabla'\cdot{\bf A} = 0,
$

    is adopted then the equation simplifies to

    $\displaystyle {\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

    $\displaystyle \frac{\partial\rho}{\partial t} +\nabla' \cdot{\bf j} = 0,
$

    where

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

    and

    $\displaystyle {\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.


next up previous
Next: Orbital Angular Momentum Up: Quantum Dynamics Previous: Flux Quantization and the
Richard Fitzpatrick 2016-01-22