The Hamiltonian of a two-state system is given by Hˆ = H11 |1ih1| + H22 |2ih2| + H12 |1ih2| + H21 |2ih1| where the states |1i, |2i form an orthonormal basis. Since Hˆ must be Hermitian, H11 and H22 are real, while the complex (off-diagonal) elements satisfy H∗ 21 = H12. Find the eigenvalues of Hˆ . Find also the eigenvectors of Hˆ (as linear combinations of the states |1i and |2i).

The Hamiltonian of a two-state system is given by Hˆ = H11 |1ih1| + H22 |2ih2| + H12 |1ih2| + H21 |2ih1| where the states |1i, |2i form an orthonormal basis. Since Hˆ must be Hermitian, H11 and H22 are real, while the complex (off-diagonal) elements satisfy H∗ 21 = H12. Find the eigenvalues of Hˆ . Find also the eigenvectors of Hˆ (as linear combinations of the states |1i and |2i).

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