
Objectives
A wave has fixed points, the nodes, in which the function
y(x) is nil and other points, the antinodes, in which the
amplitude of the oscillation is maximum. The eigenvalues constitute a
series of discrete values which can be reduced to a series of integers by
a simple transformation. The waveforms of the main nodes are sinusoidal
and the number of nodes of the string n (excluding the ends) is linked to
the order number of the corresponding mode m of the equation n = m 
1 . By inclusion of other possible schemes of connection to the
external system in addition to the case of a string with fixed ends, this
analogy provides an exhaustive explanation and clarification of the
intuitions and imagination regarding the significance of states of
particles.In fact, if we assume that it is equally possible for the
external system to have mechanical characteristics, it can be demonstrated
that the vibrations of the string are different from those which occur
when the external system is reduced to two fixed points. The order number
is dependent on the structure of the environment to which the string is
connected. Analogy with the actual state of physics may be
derived.
Contents Still in the field of
stationary vibrations in stretched cords, let us now examine the analogy
presented by Einstein and Infeld to explain the discontinuity of levels of
atomic energy. In Einstein, one can read "the number of nodes must be a
whole number and can only change occasionally. A figure of 3.756 for the
number of nodes of a stationary wave would make no sense and this implies
that the wavelength can also vary only in a discontinuous way. This
classic problem presents all the familiar characteristics of the quantum
theory". Cx x is a
string; Sy y is a
system external to the string; Mx x is endowed with movement
q is a physical
law which expresses this movement i the number of nodes of the
stationary vibratory waves is an integer
Results If, instead, one also considers
the mechanical structures of the system external to the string,
interpreting its behaviour with the laws used for the string itself, then
this strict numerability of the modes of vibration and the wave nodes no
longer holds good and, in addition, other stationary harmonic forces
appear which are mainly due to the interaction between the string and the
external system. Cases which constitute immediate examples have been
chosen here from the infinite range of possibilities and criteria which
could have been used to represent the continuum constituted by the system
external to the string. In these terms, the case of the fixed ends appears
as an extreme case, in which the flexural rigidity of the mechanical
system external to the string is considered to be infinite. Thus this
system is not affected by the dynamic state of the string, transmitted
externally by the two ends. The range of system
frequencies treated in case I is constituted by the infinite series of
positive integers. In case II, the effect of the elastic and inertial
characteristics of the system external to the string is such that the
eigenvalues are shifted from integers to values close to them but no
longer integers and other eigenvalues are added to the series
corresponding to those which the system external to the string would have
if it were possible to consider the system independently from the string.
Similarly, in case IV the frequency values are shifted to nonintegers.
Cases III and V represent confirmation that consideration of fixed
constraints tends to provide solutions with integers.
Future possible
developments We can, therefore, conclude that a logic
scheme with two values was used in the analogue of the vibrating string in
order to throw greater light on a phenomenon which is typical of quantum
mechanics. This analogy does, in fact, provide an exhaustive explanation
and clarifies the intuitions and imagination regarding the significance of
the stationary states of atomic particles. On the other hand, if we
use a logic scheme with three values, we use a theoretical approach which
is more consistent with the subject involved but the results which are
obtained are not so evident and significant, since the clear discontinuity
of the vibrating states and the quantum characterisation is less obvious.
While the two logic schemes and the two physical theories develop
differently, the conclusions are interchangeable in the case of this
problem. 