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![Consider a two-level Hamiltonian \( \hat{H} \) with orthonormal energy eigenstates \( |1\rangle \) and \( |2\rangle \). In terms of \( |1\rangle \) and \( |2\rangle \), let \( \hat{H} \) be given by
\[
\hat{H} = E_0 \left( |1\rangle \langle1| - |2\rangle \langle2| \right) + E_1 \left( |1\rangle \langle2| + |2\rangle \langle1| \right),
\]
where \( E_0 \) and \( E_1 \) are positive constants with units of energy. Diagonalize this Hamiltonian and write down its energy eigenvalues in terms of \( E_0 \) and \( E_1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2Fdf6ae8b9-4050-46d4-b348-948328bac2f0%2Foag4wo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a two-level Hamiltonian \( \hat{H} \) with orthonormal energy eigenstates \( |1\rangle \) and \( |2\rangle \). In terms of \( |1\rangle \) and \( |2\rangle \), let \( \hat{H} \) be given by
\[
\hat{H} = E_0 \left( |1\rangle \langle1| - |2\rangle \langle2| \right) + E_1 \left( |1\rangle \langle2| + |2\rangle \langle1| \right),
\]
where \( E_0 \) and \( E_1 \) are positive constants with units of energy. Diagonalize this Hamiltonian and write down its energy eigenvalues in terms of \( E_0 \) and \( E_1 \).
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Matrix representation of this "Hamiltonian" operator is given by,
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