Constrained motion on a line or on a surface
A possible objection to the assumption of a fixed three-dimensionl lattice (with dimensions of the order of the Planck length) where quantum-mechanical objects evolve, is that this assumption could be in contrast with motion on an arbitrary subset of the lattice, e.g., a line or a surface.
However, these scenarios can be easily represented in the proposed model. Consider a line, that is, a succession of lattice nodes that are described by a set of numbers {a} whose sum of squares is equal to one. We have discussed in the previous post that these numbers are the direction cosines of the normal vector to the line. In that post, we have also presented the generalized rules of motion when these node represent a potential barrier. Particle hitting such nodes experience an external force (i.e., they capture an external "boson" residing in the node) with a consequent momentum propensity change, as well as a span reset.
Now, imagine that all nodes outside the prescribed line are potential barriers. Whenever the particle tries to exit the line, it hits one such node. According to the aforementioned rules, a component of the momentum propensity is inverted, namely, that perpendicular to the line
while the parallel component
is conserved. Motion along the normal direction is thus equivalent to a one-dimensional particle-in-a-box, with a box width 2w = 1 (in lattice units). When one evaluates the PBM associated with different numbers of hitting the box, one easily realizes that the joint pdf of the quantum momentum only depends on the modulus of the perpendicular component (i.e., normal to the line) above. In fact, the argument of the sine terms deriving from the LBM exchanged is of the form
and the span differences δ are proportional to the direction cosines a.
The remaining part, i.e., the parallel component is thus 'free', that is, solely determined by the source momenta. In average , after a sufficiently long time, the normal component is null (it jumps from a stationary value to its inverse, see the post on particle in a box), and the particle is actually confined to move along the line.
Analogously, according to the span reset rule, the component of the 3D span normal to the line is inverted, while the parallel component is conserved. Consequently, the normal span is null in average, and the residual span (the parallel one) reflects the distance spanned along the line.
In summary, everything works as if particle would move on a curvilinear lattice on the line. However, note that, while standard mathematical physics describes this particle-on-a-line scenario with an actual change from Cartesian to curvilinear coordinates, in the proposed model there is no coordinate change needed nor possible, as the lattice is assumed to be fixed, and the features of QM are captured as they naturally arise from the lattice description and the reset rules.
Similar considerations apply for the case of a particle bounded to remain on a surface in a three-dimensional space.
However, these scenarios can be easily represented in the proposed model. Consider a line, that is, a succession of lattice nodes that are described by a set of numbers {a} whose sum of squares is equal to one. We have discussed in the previous post that these numbers are the direction cosines of the normal vector to the line. In that post, we have also presented the generalized rules of motion when these node represent a potential barrier. Particle hitting such nodes experience an external force (i.e., they capture an external "boson" residing in the node) with a consequent momentum propensity change, as well as a span reset.
Now, imagine that all nodes outside the prescribed line are potential barriers. Whenever the particle tries to exit the line, it hits one such node. According to the aforementioned rules, a component of the momentum propensity is inverted, namely, that perpendicular to the line
while the parallel component
is conserved. Motion along the normal direction is thus equivalent to a one-dimensional particle-in-a-box, with a box width 2w = 1 (in lattice units). When one evaluates the PBM associated with different numbers of hitting the box, one easily realizes that the joint pdf of the quantum momentum only depends on the modulus of the perpendicular component (i.e., normal to the line) above. In fact, the argument of the sine terms deriving from the LBM exchanged is of the form
and the span differences δ are proportional to the direction cosines a.
The remaining part, i.e., the parallel component is thus 'free', that is, solely determined by the source momenta. In average , after a sufficiently long time, the normal component is null (it jumps from a stationary value to its inverse, see the post on particle in a box), and the particle is actually confined to move along the line.
Analogously, according to the span reset rule, the component of the 3D span normal to the line is inverted, while the parallel component is conserved. Consequently, the normal span is null in average, and the residual span (the parallel one) reflects the distance spanned along the line.
In summary, everything works as if particle would move on a curvilinear lattice on the line. However, note that, while standard mathematical physics describes this particle-on-a-line scenario with an actual change from Cartesian to curvilinear coordinates, in the proposed model there is no coordinate change needed nor possible, as the lattice is assumed to be fixed, and the features of QM are captured as they naturally arise from the lattice description and the reset rules.
Similar considerations apply for the case of a particle bounded to remain on a surface in a three-dimensional space.
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