The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 29

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The role of quantum mechanics in signal
processing

In quantum mechanics, the dynamics of physical systems that evolve in time
are described by the Schrodinger equation U(τ)|Υ k ) = u k |Υ k ) where u k =

exp (−iω k τ). This represents the eigenvalue problem of the evolution operator
U = exp (−i ˆ τ) where ˆ is the system operator which generates the dynamics.
To secure decay to zero of u k at the infinite time, we must have Im(ω k ) < 0.
Physically, ˆ is the energy operator associated with the Hamiltonian which
is the total energy (kinetic + potential) of the whole system. For resonances
with complex energies, the operator ˆ is nonHermitean, ˆ † = ˆ . Given a
'Hamiltonian' ˆ of the studied system, direct spectral analysis deals with ex
traction of the spectral set with all the eigenfrequencies and eigenfunctions
{ω k , Υ k }. Such a task is usually accomplished by solving the Schrodinger
eigenvalue problem via, e.g., diagonalization of ˆ by employing certain basis
functions from a set which is complete, or locally complete. For practical
purposes, all one needs are the complex frequencies{ω k }and the correspond
ing complex amplitudes{d k }. By definition, the amplitudes follow from the
squared projection of the full state vector Υ k onto the initial state Φ 0 of the
system, d k = (Φ 0 |Υ k ) 2 . In particular, the absolute values{|d k |}are the inten

sity of spectral lineshapes, whereas{Re(ω k )}and{Im(ω k )}are, respectively,
equal or proportional to the positions and widths of the resonances in the
spectrum of the investigated system. Hereafter, the notation Re(z) and Im(z)
will be used to label the real and imaginary parts of the complex quantity z,
respectively. Thus, by extracting all the spectral parameters{ω k ,d k }for the

given system operator ˆ , quantum mechanics can examine the structure of
matter on any concrete level (nuclear, atomic, molecular, etc.). This is known
as the direct quantification problem.
The rationale for calling the quantity U the evolution operator can at
once be appreciated by inspecting the timedependent Schrodinger equation
(i∂/∂t)Φ(t) = ˆ Φ(t). 'Hamiltonian' operator ˆ is stationary for conservative

systems, in which case the solution Φ(t) of the Schrodinger equation can be
generated simply through the relation Φ(t) = U(t)Φ(0). In other words, Φ(t)
at an arbitrary time t > 0 is obtained by subjecting Φ(0) to the action of
the operator U(t). Therefore, given the operator ˆ , the state of the system
Φ(t) at any instant t will be known if the initial state Φ(0) = Φ(t = 0)≡Φ 0
is specified at t = 0. This is the origin of determinism of quantum mechan

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