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5.P: Exercises

  1. Demonstrate that the operators defined in Equations (427)-(429) are Hermitian, and satisfy the commutation relations (417).

     

  2. Prove the Baker-Hausdorff lemma, (447).

     

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

     

  4. 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)$ .

     

  5. 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. Determine the constant $ A$ by normalizing $ \chi$ . 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 above calculations for $ S_x$ and $ S_y$ .

     

  6. 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$ ?

     

  7. 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$ ?

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