The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 37

The numerator P K−1 (ω) and denominator Q K (ω) are an operator and a scalar
polynomial of degree K−1 and K, respectively. Hence, expression (2.22) is
a mixed operatorscalar Pade approximant for the Green operator G(ω).
Operators do not possess their direct physical meanings in quantum me
chanics. However, their scalar products taken over physical states could have
physical meaning. Hence, it is important especially for comparison with ex
perimental data, to see the physical meaning of the overlap integrals involving
certain physical states, such as Φ 0 , weighted with operators U(t) or G(ω). As
to U(t), the answer is already known from (2.17), where the observable C(t) is
the autocorrelation function given by the overlap (Φ 0

|U(t)|Φ 0 ). Likewise, the
significance of the Green operator G(ω) is in the fact that its matrix element
taken over, e.g., the initial state Φ 0 is the Green function G(ω). This Green
function is the exact frequency spectrum of the system
G(ω)≡(Φ 0 |G(ω)|Φ 0 ). (2.23)

Substituting (2.21) into the rhs of (2.23), employing (2.19) and ignoring the
ǫ−factor, we have

K

d k
ω−ω k .

G(ω) =−i (2.24)

k=1

Either by carrying out the sum over k on the rhs of (2.24), or by putting
(2.22) into (2.23) invariably yields

G(ω) =−i P K−1 (ω)

Q K (ω) .

(2.25)

This quotient is explicitly the scalar Pade approximant for the Green function
G(ω) and the same is implicitly true for the Heaviside representation (2.24).
In an alternative derivation, inserting (2.21) into the rhs of (2.23) and iden
tifying the matrix element (Φ 0 |U(t)|Φ 0 ) as C(t) from (2.18) gives

∞

dt e iωt C(t). G(ω) = (2.26)

0

This demonstrates that the Green function G(ω) is the Fourier integral of the
autocorrelation function C(t) similarly to their operatorvalued counterparts
G(ω) and U(t) from (2.20). Due to the invertability of the Fourier integrals,
U(t) and C(t) can be retrieved exactly from

G(ω) and G(ω), respectively

∞

1
2π

U(t) =

dω e −iωt G(ω)

(2.27)

0

∞

1
2π

dω e −iωt G(ω).

C(t) = (2.28)

0