Chemistry Reference
In-Depth Information
Fig. 1.2
Exponential decay
log
N(t)
Mean life
,
τ
Half life, t
1/2
1
1/2
1/e
Time
To relate the exponential decay to properties of the decaying state, the time
dependence of the wave functions of a particle at rest is shown explicitly as
w
ð
t
Þ¼
w
ð
0
Þ
e
iEt
=
h
:
ð
1
:
4
Þ
If the energy E of this state is real, the probability of finding the particle not a
function of time because
j
2
¼
w
ð
0
Þ
j
2
:
j
w
ð
t
Þ
j
ð
1
:
5
Þ
A particle described as a wave function as shown in (
1.4
) with real E does not
decay. To introduce an exponential decay of a state described by w
ð
t
Þ
, a small
imaginary part is added to the energy,
E
¼
E
0
1
2
iC
;
ð
1
:
6
Þ
1
2
where E
0
and C are real and where the factor
is chosen for convenience. With
(
1.6
), the probability becomes
j
2
¼
w
ð
0
Þ
j
2
e
Ct
=
h
:
j
w
ð
t
Þ
j
ð
1
:
7
Þ
It agrees with the exponential law (
1.2
)if
C
¼
kh
:
ð
1
:
8
Þ
With (
1.4
) and (
1.6
) the wave function of a decaying state is
w
ð
t
Þ¼
w
ð
0
Þ
e
iE
0
t
=
h
e
Ct
=
2h
:
ð
1
:
9
Þ
By the Fourier inversion of w
ð
t
Þ
to g
ð
x
Þ
, g
ð
x
Þ
is given by