Title: Critique of Phenomena of Vibrating Strings with Stationary Waves: Analogies with Certain Interpretations of Quantum.
Author: prof. Alberto Rovetta N. BC-1

Erwin Schrodinger, The image of physics, Geneva, 1952

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 eigen-values 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.

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

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 eigen-values are shifted from integers to values close to them but no longer integers and other eigen-values 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 non-integers. 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.