Phase as an alternative to wave packets
In standard Quantum Mechanics, wave packets (where the represented variable is a probability amplitude) are an essential tool to describe particles in many scenarios. In the local-realistic model that we present in this blog, we have discussed in earlier posts how to describe ensembles of particles emitted from a discrete set of posssible sources.
Particles emitted from different sources will generally have a different span when they reach the same lattice node at the same lifetime, which results in quantum forces. Yet another piece of iformation is missing so far, to describe phase that plays such a big role, notably, in wave packets.
The model assumes that, at sources, particles are attributed a further quantity, denoted as phase source ε, that is generally a rational number, together with the momentum propensity v0. We will see in a future post that the list of source-attributed quantites is not over!
The phase source plays a role in the boson dynamics. In particular, it is stored as a second "footprint", besides the span trace, to the lattice nodes visited by the particle. In analogy to the span trace λxt, we shall label the source trace as εxt, with reference to the particular lattice node (x,t). At Quantum Reset, the phase carried by the particle is exchanged with the phase trace in the lattice.
In addition, source phase affects the LMB, the momentum of the lattice bosons when they are created. The formula generalizing that discussed in this post, reads
}\Leftarrow&space;|\ell-\lambda|&space;v_Q-\left(\epsilon-\epsilon_{xt}&space;\right&space;) "\omega_{xt}^{(\ell\lambda)}\Leftarrow |\ell-\lambda| v_Q-\left(\epsilon-\epsilon_{xt} \right )")
from where the role of the phase difference between the particle and the lattice (trace) is apparent.
Analogous of standard QM wave packets can be prepared by setting a finite number Ns of sources at adjacent nodes (e.g., centered at x = 0). The source phase is set as as proportional to the distance from the center, i.e.,
=v_mx "\epsilon(x)=v_mx")
with vm a rational number between 0 and 1. The source probability describes the type of wave packet. For a "Plane wave", it can be set as
=\frac{1}{N_s} "P_0(x)=\frac{1}{N_s}")
while for a "Gaussian wave" preparation with variance (Ns-1)/4 it should be
=\frac{1}{2^{N_s-1}}{{N_s-1}\choose&space;{x+\frac{N_s-1}{2}}} "P_0(x)=\frac{1}{2^{N_s-1}}{{N_s-1}\choose {x+\frac{N_s-1}{2}}}")
The model assumes that, at sources, particles are attributed a further quantity, denoted as phase source ε, that is generally a rational number, together with the momentum propensity v0. We will see in a future post that the list of source-attributed quantites is not over!
The phase source plays a role in the boson dynamics. In particular, it is stored as a second "footprint", besides the span trace, to the lattice nodes visited by the particle. In analogy to the span trace λxt, we shall label the source trace as εxt, with reference to the particular lattice node (x,t). At Quantum Reset, the phase carried by the particle is exchanged with the phase trace in the lattice.
In addition, source phase affects the LMB, the momentum of the lattice bosons when they are created. The formula generalizing that discussed in this post, reads
from where the role of the phase difference between the particle and the lattice (trace) is apparent.
Analogous of standard QM wave packets can be prepared by setting a finite number Ns of sources at adjacent nodes (e.g., centered at x = 0). The source phase is set as as proportional to the distance from the center, i.e.,
with vm a rational number between 0 and 1. The source probability describes the type of wave packet. For a "Plane wave", it can be set as
while for a "Gaussian wave" preparation with variance (Ns-1)/4 it should be
It will be clear that in such cases the phase is responsible for the space propagation (besides its spreading) of the packet as a whole, that is, represents the group velocity.
The figure below shows the frequency of arrivals (Nt = 500, Np = 5000) obtained with the trained code for a Gaussian wave preparation (Ns = 31, equivalent to a Gaussian wave with a variance of 7.5, vm = 0.1). Also shown are the source probability density (black) and the theoretical QM density at the screen (red), both multiplied by Np. Note that the original wave packet has translated of about 50 nodes, that is, the group velocity (0.1) times the no. of iterations (500).
The figure below shows the frequency of arrivals (Nt = 500, Np = 5000) obtained with the trained code for a Gaussian wave preparation (Ns = 31, equivalent to a Gaussian wave with a variance of 7.5, vm = 0.1). Also shown are the source probability density (black) and the theoretical QM density at the screen (red), both multiplied by Np. Note that the original wave packet has translated of about 50 nodes, that is, the group velocity (0.1) times the no. of iterations (500).
Comments
Post a Comment