The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 30

ics. This mapping justifies the name of the time evolution operator for U.
Spectral problems can alternatively be solved by reliance upon another pow
erful quantummechanical formalism called the Green operator

G(u) which is

defined as the resolvent operator, G(u) = (u1−U) −1 .
All physical systems develop in time via their correlated dynamics. Corre
lations between any two states of the system are contained in the evolution
operator U(t) which connects Φ(0) with Φ(t). Since operators are not observ
ables, i.e., they cannot be measured directly in experiments, certain related
scalar quantities are needed. For instance, the projection technique can be
used to project one state onto the other via the scalar or inner product. Thus,
to correlate Φ(t) and Φ(0) with the ensuing scalar result, one can make the
inner product of these two states, C(t)≡(Φ(0)|Φ(t)). The obtained quan
tity C(t) is known as the autocorrelation function, because it correlates the
same system to itself via two different states Φ(t) and Φ(0). Furthermore,
using Φ(t) = UΦ(0), it follows that C(t) basically correlates the state Φ(0)
to the same vector Φ(0) 'weighted' with the evolution operator U, such that
C(t) = (Φ 0 |U(t)|Φ 0 ).
The chief working hypothesis of quantum mechanics asserts that the entire
information about any given system is ingrained in the wave function of the
system. These stationary and nonstationary wave functions are the state vec
tors Υ and Φ(t), respectively. The mathematical formula which encapsulates
this postulate is given by the requests of the existence of the global (or at least
the local) completeness relation

K

k=1 π k = 1, where π k is the projection oper

ator π k =|Υ k )(Υ k |. The term 'completeness' refers explicitly to the complete

information. To interpret the completeness relation, one resorts to the prob
abilistic meaning of quantum mechanics and all its entities. Thus, the state
vector|Υ k ) and its dual counterpart (Υ k |are the probability densities to find
the system in the state described by these wave functions. Plausibly, knowing
everything about the system under study is equivalent to achieving the max
imum probability, which is equal to 1. This is precisely what is stated by the
completeness relation in its operator form. The same statement can also be
expressed in the scalar form by putting the operator completeness relation in
a matrix element between the initial states|Φ 0 ) and its dual counterpart (Φ 0 |.
The result reads as

K
k=1 d k = 1 where d k is the scalar amplitude d k given
by d k = (Φ 0 |π k |Φ 0 ) = (Φ 0 |Υ k ) 2 which agrees with the abovequoted formula
for d k . Therefore, one of the ways of checking the completeness relation is
to verify whether the sum of all the amplitudes{d k }(1≤k≤K) is equal
to unity. It should also be noted that every form (operational or scalar) of
the completeness relation implicitly contains the whole information about the
fundamental frequencies{ω k }, since Υ k and d k correspond to ω k .
The quantities that hold the complete information, the states vectors Υ and
Φ(t), can be derived from the system operator ˆ . This, in turn, means that
the entire sought information is also present in

ˆ , or equivalently, in U(t).

U(t) itself is the major physical content of C(t).

The total evolution operator