Consider an isolated hydrogen atom.  Solving the eigenvalue equation H₀|ψ> = E|ψ> we find the eigenstate |ψ_nlm> and the associated eigenvalues E_n.  These energies E_n are in very good agreement with experimentally measured energies.  However, quantum mechanics predicts that if a system is at t = 0 in an eigenstate |ψ_n> 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 |ψ_n>, 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 |ψ_n> at t = 0. Then N(t) = N₀e^-t/τ is the number of systems still in the state |ψ_n> 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 |ψ_n> during the time interval dt for each of the N(t) systems still in the state |ψ_n>.  1/τ is the probability per unit time of leaving the unstable state |ψ_n>. 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 |ψ_n> is an eigenstate of a time-independent Hamiltonian H₀ and |ψ(0)> = |ψ_n> then
|ψ(t)> = exp(-iE_nt/ħ)|ψ_n>
and the probability of finding the system at time t in the state |ψ_n> is P_n(t) = |<ψ_n|ψ(t)>|² = 1.

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

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