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Exercises

  1. Demonstrate that the operators defined in Equations (5.11)-(5.13) are Hermitian, and satisfy the commutation relations (5.1).

  2. Prove the Baker-Campbell-Hausdorff lemma, (5.31).

  3. Let the $ \sigma_i$ , for $ i=1,3$ , be the three spin-$ 1/2$ Pauli matrices. Demonstrate that $ \sigma_j\,\sigma_k = \delta_{jk} +{\rm i}\,\epsilon_{jkl}\,\sigma_l$ , and, hence, that $ \sigma$ $ \times$   $ \sigma$ $ = 2\,{\rm i}\,$$ \sigma$ and $ \{ \sigma_i, \sigma_j \} = 2 \,\delta_{ij}$ .

  4. Find the Pauli representations of the normalized eigenstates of (a) $ S_x$ and (b) $ S_y$ for a spin-$ 1/2$ particle.

  5. Suppose that a spin-$ 1/2$ particle has a spin vector that lies in the $ x$ -$ z$ plane, making an angle $ \theta$ with the $ z$ -axis. Demonstrate that a measurement of $ S_z$ yields $ \hbar/2$ with probability $ \cos^2(\theta/2)$ , and $ -\hbar/2$ with probability $ \sin^2(\theta/2)$ .

  6. An electron is in the spin-state

    $\displaystyle \chi = A\,\left(\begin{array}{c}1-2\,{\rm i}\\ 2\end{array}\right)
$

    in the Pauli representation. (a) Determine the constant $ A$ by normalizing $ \chi$ . (b) If a measurement of $ S_z$ is made, what values will be obtained, and with what probabilities? What is the expectation value of $ S_z$ ? Repeat the previous calculations for (c) $ S_x$ and (d) $ S_y$ . [61]

  7. Consider a spin-$ 1/2$ system represented by the normalized spinor

    $\displaystyle \chi =\left(\begin{array}{c}\cos\alpha\\ \sin\alpha\,\exp(\,{\rm i}\,\beta)\end{array}\right)
$

    in the Pauli representation, where $ \alpha$ and $ \beta$ are real. What is the probability that a measurement of $ S_y$ yields $ -\hbar/2$ ? [53]

  8. An electron is at rest in an oscillating magnetic field

    $\displaystyle {\bf B} = B_0\,\cos(\omega\,t)\,{\bf e}_z,
$

    where $ B_0$ and $ \omega$ are real positive constants.
    1. Find the Hamiltonian of the system.
    2. If the electron starts in the spin-up state with respect to the $ x$ -axis, determine the spinor $ \chi(t)$ that represents the state of the system in the Pauli representation at all subsequent times.
    3. Find the probability that a measurement of $ S_x$ yields the result $ -\hbar/2$ as a function of time.
    4. What is the minimum value of $ B_0$ required to force a complete flip in $ S_x$ ?

  9. In the Schrödinger/Pauli representation, the generalization of Schrödinger's time-dependent wave equation for an electron moving in electromagnetic fields is written

    $\displaystyle {\rm i}\,\hbar\,\frac{\partial\chi}{\partial t} = \frac{1}{2\,m_e...
...A})+e\,\hbar\,\mbox{\boldmath $\sigma$}\cdot{\bf B}\right]\chi -e\,\phi\,\chi,
$

    where $ {\bf A}$ the vector potential, $ {\bf B}=\nabla\times {\bf A}$ the magnetic field-strength, $ \phi$ the scalar potential, and $ \chi$ the spinor-wavefunction. The term involving the Pauli matrices comes from the electron's intrinsic magnetic moment. (See Section 5.5.) Demonstrate that this equation can also be written

    $\displaystyle {\rm i}\,\hbar\,\frac{\partial\chi}{\partial t} = \frac{1}{2\,m_e...
...$\sigma$}\cdot (-{\rm i}\,\hbar\,\nabla+e\,{\bf A})\right]\chi -e\,\phi\,\chi.
$

    The previous expression is known as the Pauli equation [80].

  10. Let $ \sigma_1$ , $ \sigma_2$ , and $ \sigma_3$ be the Pauli matrices for a spin-$ 1$ particle.
    1. Show that

      $\displaystyle (\sigma_1)^k$ $\displaystyle = \sigma_1'$   $\displaystyle \mbox{for $k$\ even}$$\displaystyle ,$    
      $\displaystyle (\sigma_1)^k$ $\displaystyle =\sigma_1$   $\displaystyle \mbox{for $k$\ odd}$$\displaystyle ,$    

      where

      $\displaystyle \sigma_1' =\left(\! \begin{array}{ccc} 1/2, &0,&1/2\\ 0,&1,&0\\ 1/2,&0,&1/2\end{array}\!\right).$    

    2. Show that

      $\displaystyle (\sigma_2)^k$ $\displaystyle = \sigma_2'$   $\displaystyle \mbox{for $k$\ even}$$\displaystyle ,$    
      $\displaystyle (\sigma_2)^k$ $\displaystyle =\sigma_2$   $\displaystyle \mbox{for $k$\ odd}$$\displaystyle ,$    

      where

      $\displaystyle \sigma_2' =\left(\! \begin{array}{ccc} 1/2, &0,&-1/2\\ 0,&1,&0\\ -1/2,&0,&1/2\end{array}\!\right).$    

    3. Show that

      $\displaystyle (\sigma_3)^k$ $\displaystyle = \sigma_3'$   $\displaystyle \mbox{for $k$\ even}$$\displaystyle ,$    
      $\displaystyle (\sigma_3)^k$ $\displaystyle =\sigma_3$   $\displaystyle \mbox{for $k$\ odd}$$\displaystyle ,$    

      where

      $\displaystyle \sigma_3' =\left(\!\begin{array}{ccc} 1, &0,&0\\ 0,&0,&0\\ 0,&0,&1\end{array}\!\right).$    

    4. Hence, deduce that the spinor matrices for rotations through an angle $ {\mit\Delta}\varphi$ about the three Cartesian axes are

      $\displaystyle T_1({\mit\Delta}\varphi)$ $\displaystyle = \left(\!\begin{array}{ccc} c^{\,2}, &-{\rm i}\sqrt{2}\,s\,c,&-s...
...\sqrt{2}\,s\,c\\ -s^{\,2},&-{\rm i}\sqrt{2}\,s\,c,&c^{\,2}\end{array}\!\right),$    
      $\displaystyle T_2({\mit\Delta}\varphi)$ $\displaystyle = \left(\!\begin{array}{ccc} c^{\,2}, &-\sqrt{2}\,s\,c,&s^{\,2}\\...
...s^{\,2},&-\sqrt{2}\,s\,c\\ s^{\,2}&\sqrt{2}\,s\,c,&c^{\,2}\end{array}\!\right),$    
      $\displaystyle T_3({\mit\Delta}\varphi)$ $\displaystyle = \left(\!\begin{array}{ccc} \exp(-{\rm i}\,{\mit\Delta}\varphi), &0,&0\\ 0,&1,&0\\ 0,&0,&\exp(\,{\rm i}\,{\mit\Delta}\varphi)\end{array}\!\right),$    

      where $ s=\sin({\mit\Delta}\varphi/2)$ and $ c=\cos({\mit\Delta}\varphi/2)$ .

    5. Suppose that a spin-$ 1$ particle has a spin vector that lies in the $ x$ -$ z$ plane, making an angle $ \theta$ with the $ z$ -axis. Demonstrate that a measurement of $ S_z$ yields $ \hbar$ , 0 , and $ -\hbar$ with probabilities $ \cos^4(\theta/2)$ , $ 2\,\sin^2(\theta/2)\,\cos^2(\theta/2)$ , and $ \sin^4(\theta/2)$ , respectively.


next up previous
Next: Addition of Angular Momentum Up: Spin Angular Momentum Previous: Spin Greater Than One-Half
Richard Fitzpatrick 2016-01-22