Tunnelling Particles
In my 2018 ArXiv paper, several one-dimensional scenarios are simulated to corroborate the model proposed to reproduce Quantum Mechanics in a localistic and realist fashion.
Besides free particle, free faller, harmonic oscillator, and the particle-in-a-box scenarios, the last scenario of that paper is the Delta potential. In this scenario, the potential acting on particles has the shape of a Dirac delta function, that is, it is zero everywhere except for a single point, where it has an infinite value. To represent such circumstance, we have set the force function as a finite rectangular barrier,
=&space;\left\{\begin{matrix}&space;\lambda/\ell&space;&&space;x&space;\in[-\ell,0]\\&space;-\lambda/\ell&space;&&space;x&space;\in&space;[-2\ell,-\ell]\\&space;0&space;&&space;\mathrm{elsewhere}&space;\end{matrix}\right. "f(x)= \left\{\begin{matrix} \lambda/\ell & x \in[-\ell,0]\\ -\lambda/\ell & x \in [-2\ell,-\ell]\\ 0 & \mathrm{elsewhere} \end{matrix}\right.")
As further explained in the paper, the interaction with the barrier has two effects: (i) let a fraction of the particles pass, while the remaining fraction is reflected, and (ii) temporally delay those that are reflected (as clearly seen in the figure above). The second effect is equivalent to an infinity of virtual sources placed at increasing distance from the barrier on the positive x side.
In the proposed model, the same effects arises naturally as a consequence of the random motion around x = 0. In fact, particles can emerge from the potential delta with several momentum values, depending on how much time they have spent in the finite rectangular barrier. This variety of exit momenta translates into a variety of equivalent ("virtual") sources.
The proposed model was used to represent both a single-source preparation and a Gaussian-wave preparation. The results are contained in that paper and are not replicated here.
For the single-source scenario, we found a reflection ratio that equals 50% for λ = 0 and increases toward 100% as λ increases. This result seems logical because particles that are emitted with random source momentum tend to spread in all directions when they are free (λ = 0), while the presence of a barrier (λ > 0) tends to favor reflection. Standard quantum mechanics predicts this result by using the quantum propagator (shown as equations 124 and 125 in the mentioned paper), that is complex calculus.
For a "wavepacket" preparation under the form of a Gaussian wave, we have a beam propagating with a definite momentum. In the proposed model, the lattice and particle training processes make the momentum of each particle emitted converge to the 'group' momentum, that is the physical beam speed. if the beam is directed toward the barrier, the model predicts a reflection ratio that is λ = 0 (all particles pass the barrier) and increases toward 100% when λ increases. In contrast, QM finds this result by superposing two waves of different amplitudes and imposing some continuity conditions.
As shown in the paper (figures below), the results computed with the proposed model are remarkably close to the QM predictions. However, the model additionally computes particle trajectories that produce the correct tunnelling statistics.
where λ is the amplitude of the Delta potential and ℓ is an arbitrary scale. Classically, particles coming from negative x are transmitted, if λ is smaller than their kinetic energy, or reflected, if λ is larger. Conversely, quantum mechanics, as well as our proposed model, predict that some particles are transmitted even in the latter case. This behavior is often called tunnelling.
Example trajectories of tunnelling particles are visible in the figure below. Red curves are classical trajectories, the green line represents the potential barrier. Left plot is for λ > ℓ, right plot for λ < ℓ.
In the proposed model, the same effects arises naturally as a consequence of the random motion around x = 0. In fact, particles can emerge from the potential delta with several momentum values, depending on how much time they have spent in the finite rectangular barrier. This variety of exit momenta translates into a variety of equivalent ("virtual") sources.
The proposed model was used to represent both a single-source preparation and a Gaussian-wave preparation. The results are contained in that paper and are not replicated here.
For the single-source scenario, we found a reflection ratio that equals 50% for λ = 0 and increases toward 100% as λ increases. This result seems logical because particles that are emitted with random source momentum tend to spread in all directions when they are free (λ = 0), while the presence of a barrier (λ > 0) tends to favor reflection. Standard quantum mechanics predicts this result by using the quantum propagator (shown as equations 124 and 125 in the mentioned paper), that is complex calculus.
For a "wavepacket" preparation under the form of a Gaussian wave, we have a beam propagating with a definite momentum. In the proposed model, the lattice and particle training processes make the momentum of each particle emitted converge to the 'group' momentum, that is the physical beam speed. if the beam is directed toward the barrier, the model predicts a reflection ratio that is λ = 0 (all particles pass the barrier) and increases toward 100% when λ increases. In contrast, QM finds this result by superposing two waves of different amplitudes and imposing some continuity conditions.
As shown in the paper (figures below), the results computed with the proposed model are remarkably close to the QM predictions. However, the model additionally computes particle trajectories that produce the correct tunnelling statistics.
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