Symmetry and a three state system

As you will learn this semester, symmetries have deep implications for quantum mechanics. We can sometimes guess infer energy eigenvalues, and eigenstates based on symmetries alone. We can check those values and states rather than going through more complex algebra.

Consider the Hamiltonian for three state system written in the $\ket{a}$, $\ket{b}$, $\ket{c}$ basis.

\[\hat{H} \doteq \hbar \omega_0 \begin{bmatrix} 2 & 0 & \frac{1}{2} \cr 0 & 1 & 0 \cr \frac{1}{2} & 0 & 2 \end{bmatrix}\]
  1. Question: Is the $\ket{a}$, $\ket{b}$, $\ket{c}$ basis the energy basis of this Hamiltonian?
    • Discuss: How do you know?
  2. Question: Based on the symmetry of the matrix, how can you infer one of the eigenvalues of the system? Which one is it?
    • Discuss: Which eigenstate does that correspond to? What does that tell you about looking at matrix elements to determine eigenvalues/states?
  3. Question: By explicit calculation show that this eigenstate solves the energy eigenvalue problem.
  4. Question: Given that the matrix is 3 by 3 and symmetric with zeros for the matrix elements $H_{i2}$ and $H_{2i}$, it will produce 2 more eigenvalues of the form $E \pm \Delta E$. What are $E$ and $\Delta E$? You don’t need to solve the eigenvalue problem explicitly.
    • Discussion: How did your group determine $E$ and $\Delta E$? What does that tell you (in general) about eigenvalues for symmetric matrices with zeros in $H_{i2}$ and $H_{2i}$?
  5. Question: Lower energy states typically correspond to antisymmetric eigenstates and higher energy to symmetric eigenstates. Can you infer the normalized eigenstates corresponding to $E \pm \Delta E$?
    • Discussion: What might be some reasons determining the energy eigenstates is useful in QM problems?