Quantum Fundamentals: NoTerm-2023
HW 8 : Due Day 20 F 3/10

  1. Spin-1/2 Time Dependence Practice S0 4641S Two electrons are placed in a magnetic field in the \(z\)-direction. The initial state of the first electron is \(\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\ i\\ \end{pmatrix}\) and the initial state of the second electron is \(\frac{1}{2}\begin{pmatrix} \sqrt{3}\\ 1\\ \end{pmatrix}\).
    1. Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at \(t = 0\).
    2. Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at some later time \(t\).
    3. Calculate the expectation values for \(S_x\), \(S_y\), and \(S_z\) for each particle as functions of time.
    4. Are there any times when all the probabilities you have calculated are the same as they were at \(t = 0\)?
  2. Frequency S0 4641S 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 \(M\)?
  3. Magnet S0 4641S

    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?

  4. Spin Three Halves Time Dependence S0 4641S A spin-3/2 particle initially is in the state \(|\psi(0)\rangle = |\frac{1}{2}\rangle\). This particle is placed in an external magnetic field so that the Hamiltonian is proportional to the \(\hat{S}_x\) operator, \(\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix} 0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}\)
    1. Find the energy eigenvalues and energy eigenstates for the system.
    2. Find \(|\psi(t)\rangle\).
    3. List the outcomes of all possible measurements of \(S_x\) and find their probabilities. Explicitly identify any probabilities that depend on time.
    4. List the outcomes of all possible measurements of \(S_z\) and find their probabilities. Explicitly identify any probabilities that depend on time.