Quantization of energy: the particle-in-a-box case
One of the classical textbook example that is used to illustrate quantum effects is certainly the particle in a box scenario. In this idealized case, a particle confined between two infinite potential barriers may only occupy certain energy levels. People say that the energy is quantized in this case.
This scenario is very naturally represented in the local-realistic model of QM we are supporting throughout this blog. In an earlier post, we have anticipated that the key role to model the particle in a box is played by the External Reset condition, discussed here. An ER is activated whenever a particle (in an ensemble of similarly-prepared particles) captures an external "boson", i.e., experiences an external force. In our scenario, this means each time the particle hits one of the barriers. At an ER, the particle span (a memory of the distance traveled) changes sign. The effect is the same as if the particle had been emitted from a virtual source located at the other side of the barrier, which acts as a mirror. All of this was already discussed in the aforementioned post.
Now we present some results obtained by simulating the particle-in-a-box scenario. A rather detailed discussion was presented in the 2018 ArXiV extended paper, that is reproduced here. In the first part, the general settings are described.
In the second part, the simulation of a single-source preparation is described.
Finally, the preparation of a "stationary state" is described.
The latter scenario is particularly interesting, since it lets emerge the quantized energy levels predicted by the QM theory. In standard QM, these energy values are obtained by imposing that the wavefunction vanishes at the box boundaries. The well known result is that only the energy levels
(in lattice units) are admissible, with n > 0. This is exactly what is obtained by taking the hatted momenta above and evaluating the energies as v2/2, knowing that L = 2a.
Once again, the local-realistic model is able to predict a key fact of quantum mechanics without appealing to complex wavefunctions and the other standard mathematical tools. Quantization of momentum and energy naturally results from the rules of motion of particles, their interaction with the lattice, and their preparation (notably, the existence of a randomly-attributed source momentum). More precisely, these discrete positive levels emerge after sufficiently large times as a consequence of lattice and particle training.
This scenario is very naturally represented in the local-realistic model of QM we are supporting throughout this blog. In an earlier post, we have anticipated that the key role to model the particle in a box is played by the External Reset condition, discussed here. An ER is activated whenever a particle (in an ensemble of similarly-prepared particles) captures an external "boson", i.e., experiences an external force. In our scenario, this means each time the particle hits one of the barriers. At an ER, the particle span (a memory of the distance traveled) changes sign. The effect is the same as if the particle had been emitted from a virtual source located at the other side of the barrier, which acts as a mirror. All of this was already discussed in the aforementioned post.
In the second part, the simulation of a single-source preparation is described.
Finally, the preparation of a "stationary state" is described.
(in lattice units) are admissible, with n > 0. This is exactly what is obtained by taking the hatted momenta above and evaluating the energies as v2/2, knowing that L = 2a.
Once again, the local-realistic model is able to predict a key fact of quantum mechanics without appealing to complex wavefunctions and the other standard mathematical tools. Quantization of momentum and energy naturally results from the rules of motion of particles, their interaction with the lattice, and their preparation (notably, the existence of a randomly-attributed source momentum). More precisely, these discrete positive levels emerge after sufficiently large times as a consequence of lattice and particle training.
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