|1> and |2> are the only two states in the system. The |1> state is the ground state, while the |2> is the excited state. These eigenkets are orthonormal and form a complete set. The energy difference between the states is . An atom can move from the ground state to the excited state by absorbing a quantum of energy, , from the field. If the atom decays from the excited state to the ground state, then a quantum of energy, , is lost to the field. In both cases, this quantum of energy corresponds to a photon. There are so many photons in the field that a few extra or missing photons will not affect the total field.
Energy is arbitrary up to an additive constant; so, the energy of the ground state will be defined as zero. Therefore, . If the atom's hamiltonian operated on the ground or excited state, then
An arbitrary quantum state for a system is defined as the linear combination of all the possible states multiplied by a time--dependent coefficient, . One of the joys of the two--state atom is that an arbitrary state is so easily formulated:
All the spatial information lies in |1> and |2>, while the temporal information is in and . The spatial information depends on the exact nature of the state, i.e., S,P,D,F, etc.
Assume that is normalized. Multiplying by its hermitian conjugate, one obtains the equation
and are the probabilities of being in the ground state and the excited state, respectively, at time t. These are probability amplitudes. Understanding how these functions vary in time will tell us how the states vary and thus how the field is affecting the atom. This knowledge is necessary to fathom the underlying physics.