External reset in action: potential barriers
This blog is aimed at discussing how quantum mechanics can be modeled in a local and realistic way. In a recent post, we have introduced the external force fields in the (one-dimensional, so far) model. The key roles in this respect are played by the external bosons and the External reset. The latter rule is activated each time an external boson is captured (i.e., each time an external force is experienced) and resets the particle's span (a counter carried on by each particle representing the space it has traveled since the source) to its inverse.
With quadratic or other continuous potentials, where we have assumed that an external boson is captured at each iteration (a continuous effective momentum transfer conveniently incorporates the probability of actually finding an external boson), the ER rule plays no role. Also, as discussed in this post, quantum forces cannot be induced by continuous potentials only, since no span differences between successive particles of the same ensemble are possible.
However, the ER comes into play when the force field is not continuous. In this case, the effective momentum transfer f ("force") does not concern all possible lattice nodes. Consequently, the sign of the span depends on the path taken or, more precisely, on the number of external bosons captured. Therefore, even for particles emitted from a single source, different spans can be monitored at a given lattice node. As a consequence, quantum forces arise.
A good example of such discontinuous force fields is a potential barrier. In 1D, the barrier is constituted by a certain node xB. When a particle hits the barrier, it experiences an external force that depends on its momentum propensity,
The momentum is thus basically inverted. Particles that tended to enter the barrier are thus repulsed in the opposite direction. The "barrier" actually acts as a barrier, that is, will be penetrated with low probability as the momentum propensity points now out of the barrier.
For the ER mechanism, the span also changes at each hit as it is inverted. The barrier node acts as a mirror forming a virtual image of the actual source. If the true source was at x0, the span before hitting is xB-x0; after hitting it jumps to x0-xB, which is equivalent to the span from a virtual source placed at 2xB-x0. In future posts we will generalize such findings to the 3D case.
We are now ready to discuss the textbook situation where there are two such barriers at x = ±ℓ, which is commonly referred to as particle in a box. This scenario will be the subject of a future post.
However, the ER comes into play when the force field is not continuous. In this case, the effective momentum transfer f ("force") does not concern all possible lattice nodes. Consequently, the sign of the span depends on the path taken or, more precisely, on the number of external bosons captured. Therefore, even for particles emitted from a single source, different spans can be monitored at a given lattice node. As a consequence, quantum forces arise.
A good example of such discontinuous force fields is a potential barrier. In 1D, the barrier is constituted by a certain node xB. When a particle hits the barrier, it experiences an external force that depends on its momentum propensity,
The momentum is thus basically inverted. Particles that tended to enter the barrier are thus repulsed in the opposite direction. The "barrier" actually acts as a barrier, that is, will be penetrated with low probability as the momentum propensity points now out of the barrier.
For the ER mechanism, the span also changes at each hit as it is inverted. The barrier node acts as a mirror forming a virtual image of the actual source. If the true source was at x0, the span before hitting is xB-x0; after hitting it jumps to x0-xB, which is equivalent to the span from a virtual source placed at 2xB-x0. In future posts we will generalize such findings to the 3D case.
We are now ready to discuss the textbook situation where there are two such barriers at x = ±ℓ, which is commonly referred to as particle in a box. This scenario will be the subject of a future post.
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