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
   $$\chi = A\,\left(\begin{array}{c}1-2\,{\rm i}\\ 2\end{array}\right)$$
   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
   $$\chi =\left(\begin{array}{c}\cos\alpha\\ \sin\alpha\,\exp(\,{\rm i}\,\beta)\end{array}\right)$$
   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
   $${\bf B} = B_0\,\cos(\omega\,t)\,{\bf e}_z,$$
   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
   $${\rm i}\,\hbar\,\frac{\partial\chi}{\partial t} = \frac{1}{2\,m_e... ...A})+e\,\hbar\,\mbox{\boldmath $\sigma$}\cdot{\bf B}\right]\chi -e\,\phi\,\chi,$$
   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
   $${\rm i}\,\hbar\,\frac{\partial\chi}{\partial t} = \frac{1}{2\,m_e... ...$\sigma$}\cdot (-{\rm i}\,\hbar\,\nabla+e\,{\bf A})\right]\chi -e\,\phi\,\chi.$$
   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

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

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

       
       
       where

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

       
       

    2. Show that

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

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

       
       
       where

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

       
       

    3. Show that

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

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

       
       
       where

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

       
       

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

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

       $$T_2({\mit\Delta}\varphi) = \left(\!\begin{array}{ccc} c^{\,2}, &-\sqrt{2}\,s\,c,&s^{\,2}\\... ...s^{\,2},&-\sqrt{2}\,s\,c\\ s^{\,2}&\sqrt{2}\,s\,c,&c^{\,2}\end{array}\!\right),$$

       $$T_3({\mit\Delta}\varphi) = \left(\!\begin{array}{ccc} \exp(-{\rm i}\,{\mit\Delta}\varphi), &0,&0\\ 0,&1,&0\\ 0,&0,&\exp(\,{\rm i}\,{\mit\Delta}\varphi)\end{array}\!\right),$$

       
       
       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.