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
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