Quantum Fundamentals: Winter-2026
HW 8: Due W4 D5

1. Commute 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: \[\hat A\doteq \begin{pmatrix} a_1&0&0\\ 0&a_2&0\\ 0&0&a_3 \end{pmatrix}\qquad \hat B\doteq \begin{pmatrix} b_1&0&0\\ 0&0&b_2\\ 0&b_2&0 \end{pmatrix} \] where all the matrix elements are real.
   1. Do the operators \(\hat A\) and \(\hat B\) commute?
   2. Find the eigenvalues and normalized eigenvectors of \(\hat A\) and of \(\hat B\).
   3. Assume the system is initially in the state \(\vert 2\rangle\). Then the observable corresponding to the operator \(\hat B\) is measured. What are the possible measurement values, and what is the probability of obtaining each value? After this measurement, the observable corresponding to the operator \(A\) is measured. What are the possible measurement values, and what is the probability of obtaining each value?
   4. Interpret the Mathematical Model: How does the commutator \([\hat A,\hat B]\) found in part (1) help you understand the measurement outcomes and probabilities in part (3)?
2. General State

   Use a New Representation: Consider a quantum system with an observable \(A\) that has three possible measurement results: \(a_1\), \(a_2\), and \(a_3\). States \(\left|{a_1}\right\rangle \), \(\left|{a_2}\right\rangle \), and \(\left|{a_3}\right\rangle \) are eigenstates of the operator \(\hat{A}\) corresponding to these possible measurement results.

   1. Using matrix notation, express the states \(\left|{a_1}\right\rangle \), \(\left|{a_2}\right\rangle \), and \(\left|{a_3}\right\rangle \) in the basis formed by these three eigenstates themselves.

   2. The system is prepared in the state:

      \[\left|{\psi_b}\right\rangle = N\left(1\left|{a_1}\right\rangle -2\left|{a_2}\right\rangle +5\left|{a_3}\right\rangle \right)\]

      1. Staying in bra-ket notation, find the normalization constant.
      2. Calculate the probabilities of all possible measurement values when measuring the observable \(A\). Check “beasts.”

   3. In a different experiment, the system is prepared in the state:

      \[\left|{\psi_c}\right\rangle = N\left(2\left|{a_1}\right\rangle +3i\left|{a_2}\right\rangle \right)\]

      1. Find the normalization constant and write this state in matrix notation in the basis \(\{\left|{a_1}\right\rangle ,\left|{a_2}\right\rangle ,\left|{a_3}\right\rangle \}\).
      2. Calculate the probabilities of all possible measurement values when measuring the observable \(A\). Check “beasts”.

3. Spin Uncertainty 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. In the state \(\left|{-1}\right\rangle _y\), calculate the expectation values and uncertainties for measurements of \(S_x\), \(S_y\), and \(S_z\).