5.P: Exercises
- Page ID
- 1212
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- Demonstrate that the operators defined in Equations \ref{427}-\ref{429} are Hermitian, and satisfy the commutation relations \ref{417}.
- Prove the Baker-Hausdorff lemma, \ref{447}.
- 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 \( \sin^2(\theta/2)\) .
- An electron is in the spin-state 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 in the Pauli representation, where \( \beta\) are real. What is the probability that a measurement of \( -\hbar/2\) ?
- An electron is at rest in an oscillating magnetic field where \( \omega\) are real positive constants.
- Find the Hamiltonian of the system.
- 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.
- Find the probability that a measurement of \( -\hbar/2\) as a function of time.
- What is the minimum value of \( S_x\) ?
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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