series expansion. The matrix is called the time evolution matrix (which comprises the matrix elements of the corresponding time evolution operator ). It is easily proved that is unitary, meaning that . It can be shown that where When one changes the basis to the eigenvectors of the Hamiltonian, in other words, if the basis states are chosen to be the eigenvectors, then and and so the Hamiltonian is diagonal, i.e. and is of the form, Now, the unitary time evolution operator is easily seen to be given by: The factor merely contributes to the overall phase of the operator, and can usually be ignored to yield a new time evolution operator that is physically indistinguishable from the original operator. Moreover, any perturbation to the system (which will be of the same form as the Hamiltonian) can be added to the system in the eigenbasis of the unperturbed Hamiltonian and analysed in the same way as above. Therefore, for any perturbation the new eigenvectors of the perturbed system can be solved for exactly, as mentioned in the introduction. Rabi formula for a static perturbation Suppose that the system starts in one of the basis states at , say so that , and we are interested in the probability of occupation of each of the basis states as a function of time when is the time-independent Hamiltonian. The probability of occupation of state is . In the case of the starting state, , and from above, Hence, Obviously, due to the initial condition. The frequency is called the generalised Rabi frequency, is called the Rabi frequency, and is called the detuning. At zero detuning, , i.e., there is Rabi flopping from guaranteed occupation of state 1, to guaranteed occupation of state 2, and back to state 1, etc., with frequency . As the detuning is increased away from zero, the frequency of the flopping increases (to ) and the amplitude of exciting the electron decreases to . For time dependent Hamiltonians induced by light waves, see the articles on Rabi cycle and rotating wave approximation. Some important two-state systems Precession in a field Consider the case of a spin-1/2 particle in a magnetic field . The interaction Hamiltonian for this system is where is the magnitude of the particle's magnetic moment and is the vector of Pauli matrices. Solving the time dependent Schrödinger equation yields where and . Physically, this corresponds