A first challenge to quantum superposition
In this post, I have introduced how a genuine quantum phenomenon, the two-slit experiment, is represented in the proposed local-realistic model for quantum mechanics.
The QM mathematical machinery finds the good result by adding together ("superimposing") two different quantum states, corresponding to particles passing through either of the two slits. This is an application of the fundamental principle known as quantum superposition. Each of the two states is described by a complex wavefunction, say, ψ1 for the state where the particle passes through the first slit and ψ2 for the state where the particle passes through the second slit. if the two slits are equally probable, the wavefunctions read (in lattice units, where m → 1, ℏ → 1/π):
^{\frac{1}{2}}&space;e^{-\displaystyle\frac{\pi(x-x_k)^2}{2\iota&space;t}} "\psi_k =\left(\frac{1}{2\iota t} \right )^{\frac{1}{2}} e^{-\displaystyle\frac{\pi(x-x_k)^2}{2\iota t}}")
where k stands for 1, resp., 2, and xk is the k-th source position.
According to the superposition principle, the wavefunction of the state where the particle can pass through both slits is given by ψ = ψ1+ψ2. The probability is obtained by squaring the modulus of the wavefunction of the superimposed state (the Born rule), as
=\frac{1+\cos\left(\displaystyle\frac{2\pi\delta&space;x}{t}\right)}{2t} "P(x)=\frac{1+\cos\left(\displaystyle\frac{2\pi\delta x}{t}\right)}{2t}")
where δ = |x1-x2|.
In the orthodox, or Copenhagen, interpretation of QM, the superimposed state and its wavefunction represents a single quantum system, that is, a single particle in the two-slit scenario.
The proposed model is grounded on a different interpretation. There is no complex wavefunction and thus no 'state' mathematically defined. Particles follow real trajectories that are stochastically determined by the rules of motion. Probability densities can be directly evaluated from these rules of motion, without appealing to the QM machinery. As shown in this and this post, the interference pattern of the two-slit scenario and, of course, any other superposition phenomenon, is the result of a progressive training of the bosons created at lattice nodes and carried by the flying particles, after many particle emissions. Of course, these particle emissions must share the same characteristics, e.g., at each emission the source ('slit') is determined randomly according to the same probability distribution.
In this respect, the proposed model share the view with what is called ensemble interpretation (EI) of quantum mechanics. According to the EI, a state (although still represented by a complex wavefunction) represents not a single particle but an ensemble of particles. An example of an ensemble is composed by preparing and observing many copies of one and the same quantum system. An ensemble is not, for example, a single preparation and observation of one simultaneous set of particles. In other terms, EI is concerned with the ensemble collection of a virtual set of experimental preparations repeated many times.
Exactly what the proposed model does. Is it therefore the same as the ensemble interpretation? Of course not since, with respect to the EI, my model does not only compute ensemble probability densities, but also associates real trajectories to each repetition within an ensemble. In a way, different trajectories within the same ensemble interact among each others through the mediation of the lattice and via the creation/evolution of the "bosons". The superposition emerges from such a mediated - thus purely local - interaction.
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| Illustration by user Sakurambo, Wikimedia Commons, https://commons.wikimedia.org/wiki/File:Young_diffraction.svg |
The QM mathematical machinery finds the good result by adding together ("superimposing") two different quantum states, corresponding to particles passing through either of the two slits. This is an application of the fundamental principle known as quantum superposition. Each of the two states is described by a complex wavefunction, say, ψ1 for the state where the particle passes through the first slit and ψ2 for the state where the particle passes through the second slit. if the two slits are equally probable, the wavefunctions read (in lattice units, where m → 1, ℏ → 1/π):
where k stands for 1, resp., 2, and xk is the k-th source position.
According to the superposition principle, the wavefunction of the state where the particle can pass through both slits is given by ψ = ψ1+ψ2. The probability is obtained by squaring the modulus of the wavefunction of the superimposed state (the Born rule), as
where δ = |x1-x2|.
In the orthodox, or Copenhagen, interpretation of QM, the superimposed state and its wavefunction represents a single quantum system, that is, a single particle in the two-slit scenario.
The proposed model is grounded on a different interpretation. There is no complex wavefunction and thus no 'state' mathematically defined. Particles follow real trajectories that are stochastically determined by the rules of motion. Probability densities can be directly evaluated from these rules of motion, without appealing to the QM machinery. As shown in this and this post, the interference pattern of the two-slit scenario and, of course, any other superposition phenomenon, is the result of a progressive training of the bosons created at lattice nodes and carried by the flying particles, after many particle emissions. Of course, these particle emissions must share the same characteristics, e.g., at each emission the source ('slit') is determined randomly according to the same probability distribution.
In this respect, the proposed model share the view with what is called ensemble interpretation (EI) of quantum mechanics. According to the EI, a state (although still represented by a complex wavefunction) represents not a single particle but an ensemble of particles. An example of an ensemble is composed by preparing and observing many copies of one and the same quantum system. An ensemble is not, for example, a single preparation and observation of one simultaneous set of particles. In other terms, EI is concerned with the ensemble collection of a virtual set of experimental preparations repeated many times.
Exactly what the proposed model does. Is it therefore the same as the ensemble interpretation? Of course not since, with respect to the EI, my model does not only compute ensemble probability densities, but also associates real trajectories to each repetition within an ensemble. In a way, different trajectories within the same ensemble interact among each others through the mediation of the lattice and via the creation/evolution of the "bosons". The superposition emerges from such a mediated - thus purely local - interaction.
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