Schrödinger and Born retrieved
Quantum Mechanics theory describes systems with wave functions and their evolution with the Schrödinger equation. Probabilities of the outcomes of measurements on a quantum system are given by the Born rule. In particular, the probability of a single particle to be in a certain position is obtained by squaring the modulus of the wave function (which is a function of position).
In our local-realistic model of quantum mechanics, there is no Hilbert space, no complex numbers, no wave functions. As we have illustrated in previous posts, probability density functions (themselves, large-time approximations of probability mass functions) are obtained directly from the stochastic rules of motion assumed. For example, we have seen in this post that, for a free particle, the pdf of position is obtained as
=\frac{1+\sum_{\ell,\lambda}\sqrt{P_0^{(\ell)}P_0^{(\lambda)}}\cos{\left(\pi\delta^{(\ell\lambda)}\frac{x-(x_0^{(\ell)}+x_0^{(\lambda)})/2)}{t}-\epsilon^{(\ell\lambda)}\right)}}{2t} "\rho(x;t)=\frac{1+\sum_{\ell,\lambda}\sqrt{P_0^{(\ell)}P_0^{(\lambda)}}\cos{\left(\pi\delta^{(\ell\lambda)}\frac{x-(x_0^{(\ell)}+x_0^{(\lambda)})/2)}{t}-\epsilon^{(\ell\lambda)}\right)}}{2t}")
where the indexes l and λ represent any two of the possible source, x0 their locations, P0 their probability, δ their relative distance (reflected by the difference between span and trace), and ε the difference in their phase. Again, this result is obtained just by applying the compound probability rules to the stochastic variable "position" x once it is expressed as a function - by virtue of the rules of motion - of the initial (source) position and momentum.
Now, the Schrödinger equation and the Born rule can be retrieved from this pdf. And not vice versa! That is explained in detail in the ArXiV 2017 paper, as reproduced here:
Therefore, the QM action (and consequently Schrödinger and Born) is retrieved from the local-realistic rules of motion. Of course, this derivation is only valid for the case of a free particle, but it can be extended rather smoothly to more general scenarios. That will be the subject of future posts.
In our local-realistic model of quantum mechanics, there is no Hilbert space, no complex numbers, no wave functions. As we have illustrated in previous posts, probability density functions (themselves, large-time approximations of probability mass functions) are obtained directly from the stochastic rules of motion assumed. For example, we have seen in this post that, for a free particle, the pdf of position is obtained as
where the indexes l and λ represent any two of the possible source, x0 their locations, P0 their probability, δ their relative distance (reflected by the difference between span and trace), and ε the difference in their phase. Again, this result is obtained just by applying the compound probability rules to the stochastic variable "position" x once it is expressed as a function - by virtue of the rules of motion - of the initial (source) position and momentum.
Now, the Schrödinger equation and the Born rule can be retrieved from this pdf. And not vice versa! That is explained in detail in the ArXiV 2017 paper, as reproduced here:
Therefore, the QM action (and consequently Schrödinger and Born) is retrieved from the local-realistic rules of motion. Of course, this derivation is only valid for the case of a free particle, but it can be extended rather smoothly to more general scenarios. That will be the subject of future posts.
So it is and much more advanced !
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