Particle on a Ring and the Quantization of Angular Momentum
In this blog, we are supporting the view that a localist and realist interpretation ("model") of quantum mechanics is indeed possible. This view has many benefits, as we have discussed in several previous posts. For instance, no need for such controversial concepts as wafunction collapse (no need for wavefunctions at all).
The proposed model has been published here and here. The key feature of the model is motion on a discrete spatio-temporal lattice. In recent posts, we have presented the local-realistic rules that govern motion and interactions of particles in a three-dimensional spatial lattice. Moreover, we have disucssed how geometric constraints (such as motion on a line or surface) can be embodied while still keeping the universal structure of the lattice.
In this post we want to discuss one particular application of constrained motion, that is, the scenario commonly known as "particle on a ring". In this scenario, a particle is confined on a set of lattice nodes that form a circle of radius r. In reality, r is an integer and the "ring" is the digital-geometry analogue of an ordinary (Euclidean) ring. For definiteness, we shall consider that the ring lays on the x1-x2 plane, so that x3 = 0.
For what discussed in this post, the action of span reset rule makes the situation as if the particle was moving on a curvilinear one-dimensional lattice, where the relevant dimension is rφ, with φ the standard polar angle measured clockwise from the (0,r) point in the x1-x2 plane. Thus the motion can be equivalently described as a free periodic motion on a 1D lattice described by the coordinate φ. The expected law of motion is fully determined by the momentum propensity of this equivalent 1D motion, that we denote as vQφ. The resulting peripheral span is denoted by ℓφ.
In this preprint and in this other one (v1), I have shown the results of the model obtained by running the 1D code presented here. These results have been compared with those of quantum mechanics obtained by applying the propagator for the considered scenario, which is equivalent to that of a 1D free motion with an inifinity of equally-probable virtual sources, spatially seperated by a distance 2𝜋r. Indeed, in the proposed model quantum forces arise aven in the presence of a single physical source, because a given node on the ring can be visited by particles having looped through the ring a different number of times, thus with a peripheral span that may differ for a multiple of the ring circumference.
The scenario considered here is that of "stationary state" preparation. Its description is reproduced here below.
If we define the angular momentum propensity as J3 = r vQφ, its expected value clearly tends to the quantized values ±n/𝜋. In physical units, these values correspond to the values 0, ±ℏ, ±2ℏ, etc. predicted by quantum mechanics as the result of a measurement of the angular momentum along one definite axis (x3 in our case).
As it was the case with momentum and energy, we thus retrieve one very peculiar quantum behavior without appealing to the quantum mathematic formalism - just using the motion rules proposed. More investigations about angular momentum and spin will follow in future posts.
The proposed model has been published here and here. The key feature of the model is motion on a discrete spatio-temporal lattice. In recent posts, we have presented the local-realistic rules that govern motion and interactions of particles in a three-dimensional spatial lattice. Moreover, we have disucssed how geometric constraints (such as motion on a line or surface) can be embodied while still keeping the universal structure of the lattice.
In this post we want to discuss one particular application of constrained motion, that is, the scenario commonly known as "particle on a ring". In this scenario, a particle is confined on a set of lattice nodes that form a circle of radius r. In reality, r is an integer and the "ring" is the digital-geometry analogue of an ordinary (Euclidean) ring. For definiteness, we shall consider that the ring lays on the x1-x2 plane, so that x3 = 0.
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| The discrete circle (using an analytical definition) of radius r = 20 |
For what discussed in this post, the action of span reset rule makes the situation as if the particle was moving on a curvilinear one-dimensional lattice, where the relevant dimension is rφ, with φ the standard polar angle measured clockwise from the (0,r) point in the x1-x2 plane. Thus the motion can be equivalently described as a free periodic motion on a 1D lattice described by the coordinate φ. The expected law of motion is fully determined by the momentum propensity of this equivalent 1D motion, that we denote as vQφ. The resulting peripheral span is denoted by ℓφ.
In this preprint and in this other one (v1), I have shown the results of the model obtained by running the 1D code presented here. These results have been compared with those of quantum mechanics obtained by applying the propagator for the considered scenario, which is equivalent to that of a 1D free motion with an inifinity of equally-probable virtual sources, spatially seperated by a distance 2𝜋r. Indeed, in the proposed model quantum forces arise aven in the presence of a single physical source, because a given node on the ring can be visited by particles having looped through the ring a different number of times, thus with a peripheral span that may differ for a multiple of the ring circumference.
The scenario considered here is that of "stationary state" preparation. Its description is reproduced here below.
If we define the angular momentum propensity as J3 = r vQφ, its expected value clearly tends to the quantized values ±n/𝜋. In physical units, these values correspond to the values 0, ±ℏ, ±2ℏ, etc. predicted by quantum mechanics as the result of a measurement of the angular momentum along one definite axis (x3 in our case).
As it was the case with momentum and energy, we thus retrieve one very peculiar quantum behavior without appealing to the quantum mathematic formalism - just using the motion rules proposed. More investigations about angular momentum and spin will follow in future posts.
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