The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 33

The mechanism of creating and keeping the whole information about the
system through the state vector|Φ(t) from (2.4) is rooted in the dynamics
generator U(t) or the time evolution operator. This is also called the time
relaxation operator which is given by

U(t) = e −i Ωt

U(t)|Φ 0 ).

∴ |Φ(t)) = (2.5)

According to (2.4) and (2.5), if ˆ and |Φ 0 ) are given, the determinism of
quantum mechanics predicts that the state |Φ(t)) of the entire system will
be completely known. Stated equivalently, the full information which could
conceivably be obtained about any given system under investigation will be
available, provided that one knows how to extract such information from the
time evolution operator (2.5).
As mentioned, the stationary state|Υ k ) of any studied system satisfies the
time independent Schrodinger equation

ˆ |Υ k ) = ω k

|Υ k ) 1≤k≤K. (2.6)

The solution of this equation can be found with the help of the expansion
method by which|Υ k ) is developed in terms of a selected basis set functions.
This is followed by computation of the matrix elements of ˆ with these basis
functions. Such a procedure converts (2.6) into a system of linear equations
that are solved to obtain the K eigenvalues{ω k }. Substitution of the obtained
eigenvalues into the same system of linear equations yields the K eigenstates
{|Υ k )}. Thus, even for a direct problem (2.6) these brief outlines indicate that
the explicit knowledge of ˆ is not necessary. As seen, all one needs are the
matrix elements of ˆ , and not this operator itself. Note that hereafter K is a
nonnegative integer which is the total number of the genuine physical states of
the considered system. Each given ˆ possesses a welldefined and fixed num
ber K which is usually unknown prior to analysis. However, an adequately
designed analysis and its computer implementation ought to be capable of
finding the unique value of K, since surmising this number would be unac
ceptable for any valuable practical purpose. Any guessing of K inevitably
leads to either overestimation or underestimation of the true number of
physical states. Thus, in overestimation, spurious unphysical states that are
forbidden to the investigated system would be detected. In underestimation,
some of the genuine states would be missed. As such, both overestimation
and underestimation are anathema to a proper theory as well as to practice.
Of course, the same criticism against guessing K applies also to the corre
sponding inverse problem, such as harmonic inversion for quantification in
MRS. Specifically, in the quantification problem, ˆ is completely unknown.
Nevertheless, this is not an obstacle, since only the matrix elements of ˆ are
required to solve (2.6). These matrix elements, C(t) = (Φ 0 |exp (−i ˆ t)|Φ 0 ),
are recognized as quantummechanical autocorrelation functions C(t). It is
remarkable that precisely these autocorrelation functions are measured di
rectly in experiments as free induction decay curves or time signals c(t), as