Subatomic Physics

100 The Subatomic Zoo
It agrees with the decay law (Eq. (5.33)) if
Γ=λ�. (5.38)
With Eqs. (5.35) and (5.36) the wave function of a decaying state is
� � � �
−iE0t −Γt
ψ(t) =ψ(0) exp exp . (5.39)
�
2�
The real part of ψ(t) is shown in Fig. 5.16 for positive times. The addition of a small
imaginary part to the energy permits a description of an exponentially decaying
state, but what does it mean? The energy is an observable; does an imaginary
component make sense? To find out we note that ψ(t) in Eq. (5.39) is a function
of time. What is the probability that the emitted particle has an energy E? In
other words, we would like to have the wave function as a function of energy rather
than time. A change from ψ(t) toψ(E) is effected by a Fourier transformation, a
generalization of the ordinary Fourier expansion. A short and readable introduction
is given by Mathews and Walker; (22) here we present only the essential equations.
Consider a function f(t). Under rather general conditions it can be expressed as an
integral,
f(t) =(2π) −1/2
� +∞
−∞
dω g(ω) exp(−iωt). (5.40)
The expansion coefficient in the ordinary Fourier series has become a function g(ω).
Inversion of Eq. (5.40) gives
g(ω) =(2π) −1/2
� +∞
−∞
dtf(t)exp(+iωt). (5.41)
The variables t and ω are chosen so that the product ωt is dimensionless; otherwise
exp (iωt) does not make sense. Thus t and ω can be time and frequency
or coordinate and wave number. We now set f(t) in Eq. (5.41) equal to ψ(t),
Eq. (5.39). If the decay starts at the time t = 0, the lower limit on the integral can
be set equal to zero, and g(ω) becomes
or
g(ω) =(2π) −1/2 ψ(0)
� ∞
0
� �
dt exp +i ω − E0
� � �
t exp −
�
Γt
�
2�
(5.42)
g(ω) = ψ(0)
(2π) 1/2
i�
. (5.43)
(�ω − E0)+iΓ/2
22 Mathews and Walker, Chapter 4. Short tables of Fourier transforms are given in the Standard
Mathematical Tables, Chemical Rubber Co., Cleveland, Ohio. Extensive tables can be found in
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms,
McGraw-Hill, New York, 1954.