Preface to the lecture, 1

88

 

time dependent Schrodinger equation

 

 

Fig. 5.7:  Derivation of the time dependent Schrodinger equation 

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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):

 

 

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90

 

time independent Schrodinger equation

 

 

Fig. 5.8: Derivation of the time independent Schrodinger equation

 

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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.

 

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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.

 

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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.

 

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