3.9 Spectrum and Autocorrelation Function
101
The autocorrelation function is then
A(t) =
i
|c i |
2 e
iE i t/
.
(3.83)
The energy spectrum of Ψ is given by the eigenenergies E i and by the associated
probabilities |c i |
2 . If we want to express it as a frequency distribution, we make use
of δ functions:
S(ω) =
i
|c i |
2
δ(ω − E i /) .
(3.84)
It is interesting to realize that the spectrum is connected with the autocorrelation
function by a Fourier transform:
˜
A(ω) = (2π)
−1/2
i
|c i |
2
+∞
−∞
e
i(E i /−ω)t dt =
= (2π)
1/2
i
|c i |
2
δ(E i / − ω) = (2π)
1/2 S(ω) .
(3.85)
The vice versa is of course true:
˜
S(t) = (2π)
−1/2
i
|c i |
2
+∞
−∞
δ(ω − E i /)e
−iωt dω =
= (2π)
−1/2
i
|c i |
2 e
−iE i t/
= (2π)
−1/2 A(−t) .
(3.86)
So, the energy spectrum and the autocorrelation function in principle contain the
same information.
The eigenstates may belong to a continuum spectrum, in which case the expansion
(3.81) is replaced by
|Ψ (0) =
∞
E min
c(E) |ψ E dE
(3.87)
where |ψ E denotes the state of energy E. The spectrum is then simply
S(ω) = |c(ω)|
2
.
(3.88)
Notice that |c(E)|
2 is a probability density in the energy domain, so its dimensions
are (energy)
−1 . The autocorrelation function in this case is
A(t) =
∞
E min
|c(E)|
2 e
iEt/ dE .
(3.89)
It is easy to see that the relationships (3.85) and (3.86) also hold for a continuum
spectrum.