130_notes.dvi

29.11Lifetime and Line Width

Now we have computed the lifetime of a state. For some atomic, nuclear, or particle states, this
lifetime can be very short. We know that energy conservation can be violated for short times
according to the uncertainty principle

∆E∆t≤

̄h

2

.

This means that a unstable state can have an energy width on the order of

∆E≈

̄hΓtot

2

.

We may be more quantitative. If the probability to be in the initial state is proportional toe−Γt,
then we have
|ψi(t)|^2 =e−Γt

ψi(t)∝e−Γt/^2

ψi(t)∝e−iEit/ ̄he−Γt/^2

We may take the Fourier transform of this time function to the the amplitude as a function of
frequency.

φi(ω) ∝

∫∞

0

ψi(t)eiωtdt

∝

∫∞

0

e−iEit/ ̄he−Γt/^2 eiωtdt

=

∫∞

0

e−iω^0 te−Γt/^2 eiωtdt

=

∫∞

0

ei(ω−ω^0 +i

Γ 2 )t

dt

=

[

1

i(ω−ω 0 +iΓ 2 )

ei(ω−ω^0 +i

Γ

2 )t

]∞

0

=

i

(ω−ω 0 +iΓ 2 )

We may square this to get the probability or intensity as a function ofω(and henceE= ̄hω).

Ii(ω) =|φi(ω)|^2 =

1

(ω−ω 0 )^2 +Γ

2

4

This gives the energy distribution of an unstable state. It is calledthe Breit-Wigner line shape.
It can be characterized by its Full Width at Half Maximum (FWHM) of Γ.

The Breit-Wigner will be the observed line shape as long as the densityof final states is nearly
constant over the width of the line.