Derivation and interpretation
89
5.7 Derivation of the time dependent Schrodinger equation
With the Schrodinger approach 5.15 and its derivations the derivation is continued:
The for a harmonic oscillation won relations according to equation 5.21 and 5.22 are now
inserted into equation 5.20:
This is already the sought-for Schrodinger equation, as we will see in a moment, when
we have analysed the coefficients. Because, besides equation 5.16 for the total energy W,
also the Einstein relation is valid (with the speed of light c):
(5.24)
we can replace the coefficients of the imaginary part by:
(5.25)
To achieve that equation 5.23, as required, follows from the Schrodinger equation 5.14, a
comparison of coefficients is carried out for the real part:
(5.26)
If thc kinetic energy of a particle moving with the speed v is:
(5.27)
then acccording to De Broglie this particle has the wavelength h/mv. The consideration of
the particle as matter wave demands an agreement with the wave length c/f of an electro-
magnetic wave (with the phase velocity c). The particle hence has the speed v, which
corresponds with the group velocity of the matter wave:
(5.28)
if we insert v into equation 5.27 :
(5.27*)
According to equation 5.24 on the one hand the total energy is W = w • h and on the
other hand the relation 5.28 gives
resp.:
Inserted into equation 5.27* the sought-for coefficient reads (according to eq. 5.26):
90
time independent Schrodinger equation
Fig. 5.8: Derivation of the time independent Schrodinger equation
Derivation and interpretation
91
5.8 Derivation of the time independent Schrodinger equation
The goal is reached if we are capable to fulfil the comparison of coefficients 5.26:
(5.30)
The angular frequency w is given by equation 5.18. Therefore has to be valid:
(5.31)
(5.32)
As is well-known the arithmetic and the geometric average only correspond in case the
variables are identical. In this case, as already required in equation 5.13:
(5.13)
has to hold.
From this we can draw the conclusion that the Schrodinger equation is just applicable to
the described special case (according to eq. 5.13), in which the eddy current, which tries
to increase the particle or its circular path and the potential vortex, which keeps the atoms
together and also is responsible for the stability of the elementary particles, are of
identical order of magnitude.
As a check equation 5.23 is divided by c
2
and equations 5.30 and 5.25 are inserted:
(5.14*)
This is the time dependent Schrodinger equation 5.14 resolved for
Next we replace
according to equation 5.21 with
acc. to equation 5.24:
(5.33)
If we separate the space variables
from time by the Schrodinger approach 5.15 we
obtain:
(5.34)
This quation 5.34 for the function of space coordinates
is the time independent
Schrodinger equation:
(5.35)
The solutions of this equation which fulfil all the conditions that can be asked of them (of
finiteness, steadiness, uniqueness etc.), are called eigenfunctions. The existence of
corresponding discrete values of the energy W, also called eigenvalues of the Schrodinger
equation, are the mathematical reason for the different quantum postulates.
92
Interpretation of the Schrodinger equation
Fig. 5.9: Photographs of models of the probability
densities for
different states of the hydrogen atom.
The densities are symmetrical if rotated around the
vertical axis
taken from:
: U. Gradmann/H. Wolter: Grundlagen der Atomphysik,
AVG, Frankfurt a. M. 1971, P. 190.
Derivation and interpretation
93
5.9 Interpretation of the Schrodinger equation
The interpretation of the Schrodinger equation is still disputed among physicists, because
the concept of wave packets contradicts the corpuscular nature of the elementary particles.
Further the difficulty is added that wave packets at a closer look never are connected, run
apart more or less fast, and really nothing can hinder them doing that. But for a particle the
connection represents a physical fact. Then there can be no talk of causality anymore.
The monocausal division into two different levels of reality, in a space-timely localization
and in an energetic description, does not represent a solution but rather the opposite, the
abolition of the so-called dual nature. As has been shown, the potential vortex is able to
achieve this with the help of its concentration effect.
But from the introduction of this new field phenomenon arises the necessity to interpret
the causes for the calculable and with measuring techniques testable solutions of the
Schrodinger equation in a new way. Laws of nature do not know a possibility to choose! If
they have been accepted as correct, they necessarily have to be applied.
Three hundred years ago the scholars had an argument, whether a division of physical
pheomena, like Newton had proposed it, would be allowed to afterwards investigate
them in the laboratory individually and isolated from other influences or if one better
should proceed in an integrated manner, like for instance Descartes with his cartesian
vortex theory. He imagined the celestial bodies floating in ethereal vortices.
One absolutely was aware that the whole had to be more than the sum of every single
realizato n , but the since Demokrit discussed vortex idea had to make room for the
overwhelming successes of the method of Newton. And this idea after 2100 years was
stamped, to in the meantime almost have fallen into oblivion.
Today, where this recipe for success in many areas already hits the limits of the physical
possibilities, we should remember the teachings of the ancients and take up again the
vortex idea It of course is true that only details are calculable mathematically and that
nature, the big whole, stays incalculable, wherein problems can be seen.
If we consider the fundamental field equation 5.7, we find confirmed that actually no
mathematician is capable to give a generally valid solution for this four-dimensional
partial differential equation. Only restrictive special cases for a harmonic excitation or for
certain spatial boundary conditions are calculable. The derived Schrodinger equation is
such a case and for us particularly interesting, because it is an eigenvalue equation. The
eigenvalues describe in a mathematical manner the with measuring techniques testable
structures of the potential vortex .
Other eigenvalue equations are also derivable, like the Klein-Gordon equation or the
Lionville equation, which is applied successfully in chaos theories. So our view opens, if
chaotic systems like turbulences can be calculated as special cases of the same field
equation and should be derivable from this equation.
The in pictures recorded and published structures, which at night should have come into
being in corn fields, often look like the eigenvalues of a corresponding equation. The ripe
ears thereby lie in clean vortex structures flat on the soil. Possibly potential vortices have
charged the ears to such high field strength values that they have been pulled to the soil by
the Coulomb forces.