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Inversion of logic in Schrödinger's equation

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Schrodinger obtained his equation by starting from the particular (the wave equation of a free electron) in order to get the general (the differential equation of an electron within a potential).

The correct would be to get the general (the differential equation starting from the wave equation of a non-free electron), and the case of a free electron would be a particular form of the general equation. In another words, the correct is to start from the general, and later to get the particular.

Table of contents

1 Assumption 4 of Heuristic derivation

2 How it’s written in Resnick&Eisberg book

3 Schrödinger’s Paradox

4 See also

5 Reference

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Assumption 4 of Heuristic derivation

Consider the assumption 4 of the Heuristic derivation shown in the Wikipedia article Schrödinger equation (http://en.wikipedia.org/wiki/Schrodinger_equation):

• 4- The association of a wave (with wavefunction ψ) with any particle

According to the book Quantum Physics^{( 1 )} by Eisberg & Resnick, Schrodinger developed his equation by considering the wave equation of a free particle:

Ψ(x,t) = sen2π(x/λ – vt) ...... (1)

which he put in the following form (of an equation for a free particle):

Ψ(x,t) = cos(kx – ω.t) + sen(kx – ω.t) ...... (2)

where the constant Y, after some development, gets the values Y = ±i.

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How it’s written in Resnick&Eisberg book

Translation to English the page 176 of the book Quantum Physics by Einsberg and Resnick, where they write about the assumption 4 of the Heuristic derivation:

4. The potential energy V is in general a function of x, and possibly even of t. However, there is a special important case in which

V(x,t) = Vo .... (5.10)

This is exactly the case of a free particle, since the force that actuates on the particle is given by

F = -δV(x,t)/dx

which gives F=0 if Vo is a constant. In such case the law of Newton’s motion tell us that the momentum p of the particle will be cosntant, and also that its total energy E will be constant. We have here the situation of a free particle with constant values of λ=h/p and ν=E/h, discussed in the Chapter 13.

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Schrödinger’s Paradox

So, Schrodinger considered for the electron the equation of a free particle (not submitted to any force), and therefore within a null potential.

Well, that makes no sense, since in the hydrogen atom the electron is submitted to a force of attraction with the proton (the potential is not null).

Therefore Schrodinger actually introduced a paradox in the development of his equation.

Such paradox is eliminated only by considering a new hydrogen atom in which the electron moves through helical trajectory about the proton, and there is a force of repulsion proton-electron due to repulsive gravity, which value is equal to the attraction force, in order that it’s null the total force on the electron, as shown in the article Fundamental Requeriments for the Proposal of a New Hydrogen Atom, which is the paper number 4 of the book Quantum Ring Theory^{( 2 )}. The helical trajectory of the electron is known as zitterbewegung, and it appears in the Dirac's equation of the electron.

So, as it's null the total force on the eletron, then it makes sense to apply to the electron the equation (2) above for a free particle, as made by Schrodinger, in spite of the electron is within the electrosphere of a proton.

And by this way the paradox is eliminated

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