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**Let |a'> and |a''> be eigenstates of a Hermitian operator A with eigenvalues a' and a'' respectively. (a'≠a'') The Hamiltonian operator is given by:**

H = |a'>∂<a''| + |a''>∂<a'|

where ∂ is just a real number.

Write down the eigenstates of the hamiltonian. what are their energy eigenvalues?

H = |a'>∂<a''| + |a''>∂<a'|

where ∂ is just a real number.

Write down the eigenstates of the hamiltonian. what are their energy eigenvalues?

Was feeling a bit confused by the question at first, and was just wondering if someone could let me know if my thoughts so far are on the right track?

Essentially, I just wrote the eigenstates as the kets: |E[itex]_{a'}[/itex]> and |E[itex]_{a''}[/itex]>. In order to find the eigenvalues of the energy, I constructed the matrix:

H = [tex]

\begin{pmatrix}

0 & ∂\\

∂ & 0

\end{pmatrix}

[/tex]

so that I could use the general det(H - λI) = 0 to find the eigenvalues. is this reasoning vaguely in the correct direction?

Also, was just wondering, if in the original hamiltonian equation, i'm allowed to take the ∂ symbol out (written below) because it is just a real number?

H = ∂(|a'><a''| + |a''><a'|)