The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 32

Supposing that the beginning of counting the time from an instant t 0 is taken
to be zero (t 0 = 0), the boundary condition to (2.1) reads as|Φ(0)) =|Φ 0 ).
As before, |Φ 0 ) is the known initial state of the whole system with the dy
namics generated by the total 'Hamiltonian' ˆ . Although it is not necessary
to normalize the state vector|Φ 0 ), the associated norm ought to be finite and
nonzero

|| 2 ≡(Φ 0 ||Φ 0 |Φ 0 )≡C 0 = 0 |C 0 |<∞. (2.2)

Thus, generally, the constant C 0 is not unity and, furthermore, the norm (2.2)
could be a complex number. The norm (Φ 0 |Φ 0 ) must be nonzero for both

mathematical and physical reasons. Mathematical, because this norm is a
part of the normalized state|Φ 0 ) as a denominator of the quotient|Φ 0 )/||Φ 0

||,

which prohibits division by zero. Physical, because the norm||Φ 0 ||could be

zero provided that, e.g.,|Φ 0 ) is the zero state vector|0). Time propagation of
the state|0) will give again the same zero state which is, as such, of no interest
due to the lack of any information about the system. As mentioned earlier,
to encompass resonances that correspond to complexvalued frequencies, the
'Hamiltonian' ˆ should be a nonHermitean linear operator. To make it trans
parent, nonHermiticity of the operator ˆ will be indicated in the definition
of the scalar product as the nonHermitean symmetric inner product via

(χ|ψ) = (ψ|χ). (2.3)

As seen, no conjugation through the usual star superscript is put on either
of the two state vectors (χ| or|ψ). Such a symmetry property of the inner
product is symbolically highlighted by employing small soft round brackets
|ψ) and (χ|in lieu of the conventional Dirac notationχ|and|ψencountered
in Hermitean Hamiltonians. The customary Dirac braket symbolism and the
soft brackets are interrelated byχ|ψ=ψ ∗ |χ ∗ andχ ∗ |ψ= (χ|ψ).
As emphasized, one of the major working hypotheses of quantum mechanics
is determinism, which prescribes the time evolution of the state of the system
as follows. Whenever the wave packet|Φ(0)) of the studied arbitrary system
is wellprepared, i.e., wellcontrolled, at the initial time t = 0 and, moreover, if
a subsequent time development is dictated exclusively by the given dynamics
(with interactions being the chief constituent), then at any later instant t,
the state|Φ(t)) shall be fully determined, i.e., completely known [5]. In the
Schrodinger picture of quantum mechanics, wave functions are nonstationary,
whereas operators are stationary. Thus, for a given stationary dynamical
operator ˆ , which is exactly what is needed for the definition of a conservative
system, the solution of (2.1) becomes

|Φ(t)) = e −i Ωt |Φ 0 )

|Φ 0 )≡|Φ(0)). (2.4)

The application of an exponential operator in (2.4) can be conceived by means
of the corresponding Maclaurin series expansion [5], or alternatively, by its
cited spectral representation which is also called the spectral decomposition.