Consider an isolated hydrogen atom. Solving the eigenvalue equation H_{0}|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 H_{0},
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 H_{0} 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 N_{0} be the number of systems in the state |ψ_{n}>
at t = 0. Then N(t) = N_{0}e^{-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/τ}d 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 H_{0} and |ψ(0)> = |ψ_{n}> then

|ψ(t)> = exp(-iE_{n}t/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_{n}t).

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: H_{0} is then no longer Hermitian, and the norm of the state
vector varies with time as exp(-γ_{n}t). This is
due to the fact that the system under study is part of a larger system which is not
completely described by H_{0}.