Theoretical Proof of De Broglie’s Wave Function

2. Wave Function of Particles at Rest

Firstly, the motion of a particle should comply with the law of conservation of energy, and the probability of a particle appearing at a certain place in space should not violate the law of conservation of energy. According to the conclusions in the “Unified Theory of Forces,” we know that all “forces” originate from the energy deficit $E_{Missing}$ between materials [1]. Therefore, the distribution of particles in space will be constrained by the “$E_{Missing}$,” and they will not appear anywhere in space unless the particle obtains energy from the outside or from other internal particles.

Based on the formula for electric potential energy:

$$E_p=k{\displaystyle \frac{Qq}{r}}$$

We got,

$$r=k{\displaystyle \frac{Qq}{E_p}}$$

Based on our conclusions in the “Unified Theory of Forces,” assuming the energy deficit between the electron and proton is $E_{Missing}^0$, then before the hydrogen atom releases energy to the outside, the equilibrium radius $\overline{r}$of the electron and proton is:

$$\overline{r}=k{\displaystyle \frac{e^2}{E_{Missing}^0}}$$

We have already proven that when the distance between the electron and proton is less than $\overline{r}$, there will be a repulsive force between the electron and proton, rather than an attractive force. This is the essence of the repulsive force between all particles at close distances [5].

When the hydrogen atom releases 13.6eV of energy (ground state hydrogen atom), we simply assume that we ignore the effect of the atomic nucleus, then according to the constraints of energy conservation, the motion radius of the electron cannot be greater than:

$$r_1=k{\displaystyle \frac{e^2}{13.6\times 1.6\times {10}^{-19}}}$$

$$=8.99\times {10}^9{\displaystyle \frac{{\left(1.6\times {10}^{-19}\right)}^2}{13.6\times 1.6\times {10}^{-19}}}\approx 1.06\times {10}^{-10}m$$

Therefore, for the ground state hydrogen atom at rest, if it does not obtain external energy, the electron outside the nucleus cannot appear outside the atomic nucleus $r_1\approx 1.06\times {10}^{-10}m$.

For the moving free particle, due to the asymmetry of motion [2] and according to the “prohibition of energy arbitrage” theorem [1], we have already proven that particles will produce Lorentz contraction effects symmetrically in all directions (not just in the direction of particle motion) [6].

Therefore, for free particles, taking the ground state hydrogen atom as an example, assuming the ground state hydrogen atom moves at a speed $v$, then its distribution in space will be confined to a finite range centered on the motion trajectory of the hydrogen atom nucleus $r_v$:

$$r_v\le r_1\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}$$

Where, $r_1\approx 1.06\times {10}^{-10}m$

We have discussed in another article that matter, including photons and electrons, is composed of positive and negative charges through the method of energy deficit $E_{Missing}$ [5,6]. Therefore, for this type of free particles composed of positive and negative charges, their distribution in space is constrained by the above formula.

This is completely different from the wave function of free particles in traditional quantum mechanics (de Broglie wave function). The formula for the wave function of free particles in traditional quantum mechanics is:

$$\psi \left(\mathit{r},t\right)={\displaystyle \frac{A}{r}}{e}^{i\left(\mathit{k}·\mathit{r}-\omega t\right)}$$

According to the interpretation of the wave function formula for free particles in quantum mechanics, the probability of a particle appearing in a space perpendicular to the direction of motion within an integer wavelength $\lambda$ is the same. However, in fact, this is not consistent with all our experimental observations, nor does it conform to the basic law of conservation of energy.

We know that the wave nature of a particle only occurs after the interaction between objects, that is, wave nature needs material interaction as a medium condition. For example, diffraction, interference, double-slit interference, transmission, and reflection, etc. Therefore, before free particles interact with other materials, their spatial distribution is strictly limited, not uniformly distributed in the entire space.