The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 34

encountered throughout Nuclear Magnetic Resonance (NMR), Ion Cyclotron
Resonance Mass Spectroscopy (ICRMS), MRS, etc. This is the basis of the
earlier mentioned equivalence of fundamental importance between the auto
correlation functions and time signals, C(t) = c(t). Moreover, it is this equiv
alence which renders the inverse quantification problem exactly solvable by
numerical means. Such an accomplishment is made by reduction of the origi
nally nonlinear inverse problem of quantification to the same aboveoutlined
linear algebra of the direct eigenvalue problem (2.6).
The eigenfunctions{|Υ k )}form a basis set of complete orthonormalized
state vectors so that, up to a normalization constant, we can write

(Υ k ′ |Υ k )∝δ k,k ′ (2.7)

where δ k,k ′ is the Kronecker δ−symbol, δ k,k = 1 and δ k,k ′ = 0 (k ′ = k). Here,
||Υ k ||represents the norm of the eigenstate|Υ k )

|| 2 = (Υ k ||Υ k |Υ k ). (2.8)

Given an unnormalized total wave function| Υ k ) of the system, its normalized

counterpart|Υ k ) can be cast into the usual form

|Υ k ) = N k | Υ k ) (2.9)

where N k is a normalization constant. This latter constant can be determined
from the requirement that the flux of the particles described in terms of the
wave function Υ k is fixed to be, e.g., a unit flux

|| 2 = 1

(Υ k |Υ k ) =||Υ k ∴ (Υ k ′ |Υ k ) = δ k,k ′ . (2.10)

Inserting (2.9) into (2.10) gives the standard expression for the normalization
constant

1

N k = (2.11)

|| Υ k ||

1

∴ |Υ k ) =

|| |

Υ k ). (2.12)

|| Υ k

Employing (2.6), the CayleyHamilton theorem leads to the following impor
tant equation for any operator analytic function f( ˆ )

f( ˆ )|Υ k ) = f(ω k )|Υ k ). (2.13)

Here, the eigenfunctions|Υ k ) are the same in (2.6) and (2.13), whereas the
eigenvalues ω k and f(ω k ) are associated with the former and the latter equa
tion, respectively.
As discussed, the main working postulate of quantum mechanics is that the
full information about any studied system is contained in the set of the total
wave functions{|Υ k )}. This hypothesis coheres with (2.6), since

ˆ

stores the