Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator - Advanced Models of Neural Networks - page 225

(ii) The basis functions of the neural network are the eigenstates of the quantum
harmonic oscillator: this means that the proposed neural network can capture
the particle-wave nature of information. The input variable x is viewed not
only as a crisp value (particle equivalent) but is also distributed to the normal
modes of a wave function (wave equivalent).
(iii) The basis functions of the neural network have different frequency charac-
teristics, which means that the QHO-based neural network is suitable for
multi-resolution analysis.

12.5 Uncertainty Principles for the QHO-Based Neural

Networks

12.5.1 Uncertainty Principles for Bases and the Balian-Low

Theorem

Overcoming limitations of existing image processing approaches is closely related
to the harnessing of uncertainty principle [ 94 ]. A brief description of Heisenberg's
principle of uncertainty is first given. The position of a quantum particle, i.e. of a
particle the motion of which is subject to Schrödinger's equation is defined by x
while its momentum is denoted by p. As explained in Chap. 11 , it holds

xp„ (12.19)

where „ is Planck's constant. The interpretation of Eq. ( 12.19 )isasfollows:itis
impossible to define at a certain time instant both the position and the momentum
of the particle with arbitrary precision. When the limit imposed by Eq. ( 12.19 )is
approached, the increase of the accuracy of the position measurement (decrease of
x) implies a decrease in the accuracy of the momentum measurement (increase of
p) and vice-versa.
The uncertainty principle in harmonic analysis is a class of theorems which state
that a function and its Fourier transform cannot be both too sharply localized. The
uncertainty principle can be seen not only as a statement about the time-frequency
localization of a single function but also as a statement on the degradation of
localization when one considers successive elements of a set of basis functions [ 71 ].
This means that the elements of a basis as well as their Fourier transforms cannot
be uniformly concentrated in the time-frequency plane [ 24 , 111 ].
First, a formulation of the uncertainly principle in the case of Gabor frames will
be given, through the Balian-Low theorem. A Gabor function g nm ;n;m 2 Z is
defined as (see, for instance, Fig. 11.2 b):

g mn .x/ D e imf 0 x g.x nb 0 /; n;m; 2 Z (12.20)