to write the Hamiltonian matrix in such a general form with bra-ket-enclosed Hamiltonians, since should always equal zero and should always equal . The reason is that, in some more complex problems, the state vectors may not be eigenstates of the Hamiltonian used in the matrix. One place where this occurs is in degenerate perturbation theory, where the off-diagonal elements are nonzero until the problem is solved by diagonalization. Because of the hermiticity of the eigenvalues are real; or, rather, conversely, it is the requirement that the energies are real that implies the hermiticity of . The eigenvectors represent the stationary states, i.e., those for whom the absolute magnitude of the squares of the probability amplitudes do not change with time. Eigenvalues of the Hamiltonian The most general form of a 2×2 Hermitian matrix such as the Hamiltonian of a two-state system is given by where and are real numbers with units of energy. The allowed energy levels of the system, namely the eigenvalues of the Hamiltonian matrix, can be found in the usual way. Equivalently, this matrix can be decomposed as, Here, and are real numbers. The matrix is the 2×2 identity matrix and the matrices with are the Pauli matrices. This decomposition simplifies the analysis of the system, especially in the time-independent case, where the values of and are constants. The Hamiltonian can be further condensed as The vector is given by and is given by . This representation simplifies the analysis of the time evolution of the system and is easier to use with other specialized representations such as the Bloch sphere. If the two-state system's time-independent Hamiltonian is defined as above, then its eigenvalues are given by . Evidently, α is the average energy of the two levels, and the norm of is the splitting between them. The corresponding eigenvectors are denoted as and . Time dependence We now assume that the probability amplitudes are time-dependent, though the basis states are not. The Time-dependent Schrödinger equation states , and proceeding as before (substituting for and premultiplying by again produces a pair of coupled linear equations, but this time they are first order partial differential equations: . If is time independent there are several approaches to find the time dependence of , such as normal modes. The result is that where is the statevector at . Here the exponential of a matrix may be found from the