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5.P: Exercises

  • Page ID
    1212
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    1. Demonstrate that the operators defined in Equations \ref{427}-\ref{429} are Hermitian, and satisfy the commutation relations \ref{417}.
    2. Prove the Baker-Hausdorff lemma, \ref{447}.
    3. Find the Pauli representations of the normalized eigenstates of \( S_y\) for a spin-\( 1/2\) particle has a spin vector that lies in the \( z\) plane, making an angle \( z\) -axis. Demonstrate that a measurement of \( \hbar/2\) with probability \( -\hbar/2\) with probability \( \sin^2(\theta/2)\) .
    4. 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. Determine the constant \( A\) by normalizing \( S_z\) is made, what values will be obtained, and with what probabilities? What is the expectation value of \( S_x\) and \( 1/2\) 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 \( \beta\) are real. What is the probability that a measurement of \( -\hbar/2\) ?
    5. An electron is at rest in an oscillating magnetic field $ {\bf B} = B_0\,\cos(\omega\,t)\,{\bf e}_z,
$ where \( \omega\) 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 \( \chi(t)\) that represents the state of the system in the Pauli representation at all subsequent times.
      3. Find the probability that a measurement of \( -\hbar/2\) as a function of time.
      4. What is the minimum value of \( S_x\) ?

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 5.P: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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