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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 \( E_1 \).
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 \( E_1 \).
Expert Solution
Step 1

Given:

H^=E0(|1><1|-|2><2|)+E1(|1><2|+|2><1|)

Calculation:

Matrix representation of this "Hamiltonian" operator is given by,

H^=E0E1E1-E0Now the characteristic equation is given by,|H^-λI|=0E0-λE1E1-E0-λ=0(λ+E0)(λ-E0)-E12=0λ2-E02-E12=0λ2=(E02+E12)λ=±(E02+E12)So, λ1=(E02+E12)  and λ2=-(E02+E12)

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