5.P: Exercises

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Demonstrate that the operators defined in Equations - are Hermitian, and satisfy the commutation relations .
Prove the Baker-Hausdorff lemma, .
Find the Pauli representations of the normalized eigenstates of for a spin- particle has a spin vector that lies in the plane, making an angle -axis. Demonstrate that a measurement of with probability with probability .
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 by normalizing is made, what values will be obtained, and with what probabilities? What is the expectation value of and 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 are real. What is the probability that a measurement of ?
An electron is at rest in an oscillating magnetic field ${\bf B} = B_0\,\cos(\omega\,t)\,{\bf e}_z,$ where 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 that represents the state of the system in the Pauli representation at all subsequent times.
3. Find the probability that a measurement of as a function of time.
4. What is the minimum value of ?

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