Quantum Fundamentals: Winter-2026
HW 9: Due W5 D3

  1. Frequency Consider a two-state quantum system (i.e., a system with a two-dimensional Hilbert space) 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 represented by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} where \(c\) is real and positive. Note: Both matrices are written in the same basis.
    The initial state of the system is \(\left|{\psi(t=0)}\right\rangle =\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate of \(\hat{M}\) corresponding to the larger of the two eigenvalues of \(\hat{M}\).
    1. What is the expectation value of \(M\) as a function of time?
    2. What is the frequency of oscillation of the expectation value of \(M\)?
  2. Magnet

    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 measurement values, and what is the probability of obtaining each value?

    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 \(\hat H=\omega_0\, \hat S_y\). (You can treat \(\omega_0\) as a given parameter in your answers to the following two questions.)

      • What is the state of the system at the later time t (i.e., find \(|\psi(t)\rangle\))? Express your answer as a superposition of the \(S_z\) eigenstates \(|+\rangle_z\) and \(|-\rangle_z\).

      • At time \(t\), the observable \(S_x\) is measured, what is the probability that a value \(\hbar\)/2 will be found?

  3. Probabilities of Energy (adapted from McIntyre Problem # 3.2)

    1. Show that the probability of a measurement of the energy is time independent for a general state:

      \[\left|{\psi(t)}\right\rangle = \sum_n c_n(t) \left|{E_n}\right\rangle \]

      that evolves due to a time-independent Hamiltonian.

    2. Show that the probabilities of measurements of other observables that commute with the Hamiltonian are also time independent (neither operator has degeneracy).