Unstable States

Consider an isolated hydrogen atom.  Solving the eigenvalue equation H0|ψ> = E|ψ> we find the eigenstate |ψnlm> and the associated eigenvalues En.  These energies En are in very good agreement with experimentally measured energies.  However, quantum mechanics predicts that if a system is at t = 0 in an eigenstate |ψn> of the Hamiltonian H0, then it will remain in that eigenstate.  In experiments we observe that the excited states of the hydrogen atom decay to the ground state by emission of a photon.  They are therefore not really stationary states.
The hydrogen atom is constantly interacting with the electromagnetic field.  The Hamiltonian H0 of the isolated hydrogen atom is therefore not the Hamiltonian of the system.  The actual Hamiltonian must describe the energy of the system, i.e. the energy of the atom and the field.  While the total energy is a constant of motion, it can be lost by the hydrogen atom and be taken away by a photon.
How can we, in non-relativistic quantum mechanics, deal with the instability of states?

Experimentally one observes that for a system which at t = 0 is in an unstable state |ψn>, the probability of finding it in this state at time t is P(t) = e-t/τ.
Let N0 be the number of systems in the state |ψn> at t = 0. Then N(t) = N0e-t/τ is the number of systems still in the state |ψn> at time t.
N(t + dt) - N(t) = dN(t) = -N(t) dt/τ.
-dN(t) = N(t) dt/τ.
dt/τ is the probability of leaving the state |ψn> during the time interval dt for each of the N(t) systems still in the state |ψn>.  1/τ is the probability per unit time of leaving the unstable state |ψn>. The mean time the system remains in the unstable state is
0 t e-t/τd t/τ = τ,
where e-t/τ is the probability that the system remains in the unstable state until time t and dt/τ is the probability that it decays in the next time interval dt.

If |ψn> is an eigenstate of a time-independent Hamiltonian H0 and |ψ(0)> = |ψn> then
|ψ(t)> = exp(-iEnt/ħ)|ψn>
and the probability of finding the system at time t in the state |ψn> is Pn(t) = |<ψn|ψ(t)>|2 = 1.

What happens if we replace En by  E'n = En - iħγn/2, a complex quantity.
We then have Pn(t) = |exp(-i(En - iħγn/2)t/ħ)|2 = exp(-γnt).
The probability of finding the system in the unstable state |ψn> decreases exponentially with time, as is observed experimentally.
We can therefore take into account phenomenologically the instability of a state |ψn> whose mean lifetime is τ by adding an imaginary part to its energy and by setting γn = 1/τ.

Note: H0 is then no longer Hermitian, and the norm of the state vector varies with time as exp(-γnt).  This is due to the fact that the system under study is part of a larger system which is not completely described by H0.