3D potential barriers
Keeping on our quest for a local-realistic model for quantum mechanics (QM), we have recently discussed a set of rules for motion in three dimensions that yields the correct probability density functions as those predicted by QM. These rules include both quantum and external forces.
We shall now discuss how potential barriers (that have been introduced here for 1D motion) are described in the proposed model.
Potential barriers are represented as sets of nodes where external bosons reside (when they are captured, they immediately reform) whose momentum is a function of the particle's momentum propensity V. Each of these nodes is characterized by a parameter a = {ad}, such that
Each time a particle hits the barrier, it gains an external-boson momentum
 "f_d[n]=-2a_d\left( \sum_{d'=1}^3 a_{d'}V_{d'}[n] \right )")
The meaning of the a's is now clear: they act like direction cosines of the normal vector to the barrier. For instance, for a 1D potential barrier, like those discussed in a previous post, we retrieve the formula f = -2V there displayed. For a 45° barrier, a1 = a2 = 1/√2, thus f1 = f2 = -(V1+V2) and the new momentum propensity are V1 ⟸ -V2, V2 ⟸ -V1.
Hitting a potential barrier also resets the span of a particle. Inserting the 3D span reset rule (last equation displayed in this post) into the previous equation yields
The consequence of this reset is a generalization of the virtual mirroring effect discussed for 1D potential barriers. In fact, if the particle source was at x0, after the span reset at a node xB the particle behaves as if its source were at
 "x_0+2a \sum_{d=1}^3 a_d(x_{Bd}-x_{0d})")
For a 1D potential barrier, we retrieve that the virtual source is at 2xB-x0.
The role of potential barriers is important, not only in describing multi-dimensional "boxes" that generalize the 1D "particle-in-a-box" scenario discussed in this post, but also to explain how motion in a subset of the full 3D lattice (for example, on a line, or on a surface) can be described by the proposed model. This feature will be the subject of the next post.
We shall now discuss how potential barriers (that have been introduced here for 1D motion) are described in the proposed model.
Potential barriers are represented as sets of nodes where external bosons reside (when they are captured, they immediately reform) whose momentum is a function of the particle's momentum propensity V. Each of these nodes is characterized by a parameter a = {ad}, such that
Each time a particle hits the barrier, it gains an external-boson momentum
The meaning of the a's is now clear: they act like direction cosines of the normal vector to the barrier. For instance, for a 1D potential barrier, like those discussed in a previous post, we retrieve the formula f = -2V there displayed. For a 45° barrier, a1 = a2 = 1/√2, thus f1 = f2 = -(V1+V2) and the new momentum propensity are V1 ⟸ -V2, V2 ⟸ -V1.
Hitting a potential barrier also resets the span of a particle. Inserting the 3D span reset rule (last equation displayed in this post) into the previous equation yields
The consequence of this reset is a generalization of the virtual mirroring effect discussed for 1D potential barriers. In fact, if the particle source was at x0, after the span reset at a node xB the particle behaves as if its source were at
For a 1D potential barrier, we retrieve that the virtual source is at 2xB-x0.
The role of potential barriers is important, not only in describing multi-dimensional "boxes" that generalize the 1D "particle-in-a-box" scenario discussed in this post, but also to explain how motion in a subset of the full 3D lattice (for example, on a line, or on a surface) can be described by the proposed model. This feature will be the subject of the next post.
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