Subatomic Physics

5.7. Decays 99
Figure 5.15: Exponential decay.
Figure 5.16: Real part of the
wave function of a decaying state.
It is assumed that the decaying
state is formed at t =0.
Figure 5.15 shows log N(t) plotted against t. Half life and mean life are indicated.
In one half life, one half of all atoms present decay. The mean life is the average
time a particle exists before it decays; it is connected to λ and t 1/2 by
τ = 1
λ = t 1/2
ln 2 ∼ = 1.44t 1/2. (5.34)
To relate the exponential decay to properties of the decaying state, the time dependence
of the wave function of a particle at rest (p = 0) is shown explicitly
as
�
ψ(t) =ψ(0) exp − iEt
�
. (5.35)
�
If the energy E of this state is real, the probability of finding the particle is not a
function of time because
|ψ(t)| 2 = |ψ(0)| 2 .
A particle described as a wave function of the type of Eq. (5.35) with real E does
not decay. To introduce an exponential decay of a state described by ψ(t), a small
imaginary part is added to the energy,
where E0 and Γ are real and where the factor 1
2
Eq. (5.36), the probability becomes
E = E0 − 1
2iΓ, (5.36)
|ψ(t)| 2 = |ψ(0)| 2 exp
� −Γt
is chosen for convenience. With
�
�
. (5.37)