The true eigenstates, when they exist, don't decay. They sit and spin around in phase forever. But atomic eigenstates are not true eigenstates. The reason atomic states decay is because they are coupled to photon states, and the combined photon-atom Hamiltonian doesn't have excited atom eigenstates.
When you have an atom in a box of mirrors, there are true eigenstates of the combined photon-atom system inside the box. These are states where a quantum of energy is absorbed by the atom, remitted into the box, in a steady way, so that it is sometimes in the atom, sometimes in the photons of the box. But when you make the box big, the energy will be in the photons nearly all the time, and the atom will be in its ground state, just because there are infinitely many more photon states than atomic states. In the limit of no box, the excited atomic states are never true eigenstates, they always decay into photons irreversibly. This process was described by Fermi, and the rate of irreversible decay is given by Fermi's golden rule.
For atoms and radiation, the coupling is mostly by a term in the Hamiltonian equal to $p\cdot A(x)$, where p is the momentum of the electron and A is the vector potential at the position of the electron, plus a direct two-photon term $A(x)^2$ which you can usually ignore. You evaluate the transition by expanding A in plane waves, the coefficients of which are photon creation operators, and approximating the exponential of the X operator by the first two terms of a Taylor expansion. This is called the dipole approximation.
The resulting Hamiltonian describes transitions between the pure atom stationary states into states of the atom plus a photon, and for long times, the transitions conserve energy, so that the outgoing photon carries the energy difference that is lost by the decay. The dipole approximation is essentially exact for transitions which are dipole-allowed because the atomic motion is nonrelativistic, so that the wavelength of the light is enormous compared to the atom. The result is that there are small matrix elements for transitions between the states, accompanied by creating one photon, and these give the dipole atomic transitions. This is worked out in Sakurai's book, among others.