Quantum Fundamentals: NoTerm-2021
HW 9 : Due 3/10 Wed

1. Magnet S0 4114S

   Consider a spin-1/2 particle with a magnetic moment. At time \(t=0\), the state of the particle is \(\left|{\psi(t=0)}\right\rangle =\left|{+}\right\rangle \).

   1. If the observable \(S_x\) is measured at time \(t=0\), what are the possible results and the probabilities of those results?

   2. Instead of performing the above measurement, the system is allowed to evolve in a uniform magnetic field \(\vec{B}=B_0\, \hat y\). The Hamiltonian for a system in a uniform magnetic field \(\vec B=B_0\, \hat y\) is \(H=\omega_0\, S_y\). (You can treat \(\omega_0\) as a given parameter in your answers to the following two questions.)

      • Calculate the state of the system after a time \(t\) and represent this state using the \(S_z\) basis.
      • At time \(t\), the observable \(S_x\) is measured, what is the probability that a value \(\hbar\)/2 will be found?

2. Commute S0 4114S Consider a three-dimensional state space. In the basis defined by three orthonormal kets \(\vert 1\rangle\), \(\vert 2\rangle\), and \(\vert 3 \rangle\), the operators \(A\) and \(B\) are represented by: \[A\doteq \begin{pmatrix} a_1&0&0\\ 0&a_2&0\\ 0&0&a_3 \end{pmatrix} B\doteq \begin{pmatrix} b_1&0&0\\ 0&0&b_2\\ 0&b_2&0 \end{pmatrix} \] where all the matrix elements are real.
   1. Do the operators \(A\) and \(B\) commute?
   2. Find the eigenvalues and normalized eigenvectors of both operators.
   3. Assume the system is initially in the state \(\vert 2\rangle\). Then the observable corresponding to the operator \(B\) is measured. What are the possible results of this measurement and the probabilities of each result? After this measurement, the observable corresponding to the operator \(A\) is measured. What are the possible results of this measurement and the probabilities of each result?
   4. Interpret the Mathematical Model How are questions (a) and (c) above related?
3. Frequency S0 4114S Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable \(M\) is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the expectation value of \(M\) as a function of time? What is the frequency of oscillation of the expectation value of \(M\)?
4. Spin Uncertainty S0 4114S Consider the state \(\vert -1\rangle_y\) in a spin 1 system.
   1. Discuss the direction of the spin angular momentum for this quantum system.
   2. Calculate the expectation values and uncertainties for measurements of \(S_x\), \(S_y\), and \(S_z\).