Digital Quantum Reality

The local-realistic model for quantum mechanics that we are presenting in this blog is based on a set of rules for particles' motion and interactions in a discrete spatiotemporal lattice. In a previous post , we have introduced the 3D lattice and how the model rules (earlier described in a 1D space) are generalized to it. It is time to present a Matlab code that implements the multi-dimensional model as a program. We start from the fully trained expected-motion program , where both the lattice and the particles are " trained " and in addition we simulate the expected values of momentum and position, not their actual values (this in order to limit the computing times; for a "realistic" simulation see this code  instead). For the sake of clarity we limit the number of dimensions to two; adding the third is straightforward. The main addition to the code concern the inclusion of polarization (below noted as px, py), as we have seen in the corresponding post. %...

In the proposed local-realistic model for quantum mechanics, when the rules of motion for single particle instances are applied to an ensemble of similarly prepared particles, give rise to probability densities . In a previous post , we have discussed probability distribution for position in the case of free particles without quantum forces, that is, particles that are emitted from a single source. Illustration by user Geek3, Wikimedia Commons, https://commons.wikimedia.org/wiki/File:Hydrogen_eigenstate_n5_l2_m1.png Here we shall discuss how this probability density function changes in the presence of quantum forces. The derivation is rather straightforward. We shall use the already discussed large-time approximation and replace the position x with its average value as the argument of the probability density, with ⟨x⟩ = Vt, where we take the "trained" (i.e., steady-state) value for V. We also know from the elementary rules of motion that the momentum propensity i...

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 v 0 . 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 parti...

The main results of this post about the stochastic motion of single particles can be obtained by considering such motion - for large times - as a Brownian motion with drift , where W(t) is a Wiener process  with zero mean and unit variance. The momentum propensity V here plays the role of the drift, while the variance is given by the diffusivity coefficient D  = 1- e , where e is the energy propensity (1+ V 2 )/2. For instance, we can derive the probability density from the Fokker-Planck equation of the Brownian motion, whose solution is clearly the Gaussian distribution shown in the aforementioned post, with the initial condition w = δ( x - x 0 ). Or, we can use the properties of stochastic processes to find the probability density of the  first return time to the source, i.e., the time at which the particle comes back for the first time at x =  x 0 . For some reason, I could not find a formula for that in the case of a Wiener proce...

The rules of particles' motion in the proposed model have been described as a random walk in this post , while in this post we have evaluated the probability mass function of position for one single particle emission. Each emission is characterized by one specific value of the source momentum v 0 that, in the absence of quantum or external forces, coincides with the momentum propensity V . However, for an  ensemble  of similarly-prepared particles, the source momentum is a random variable with uniform distribution in the range from -1 to 1 (see this post ). We are now interested in evaluating the position probability mass function (pmf) for the ensemble of particles. To do so, we just need to multiply the pmf w ( x ,  V ;  t ,  x 0 ) by the source momentum probability density and integrate over all possible values, The probability density function (pdf) of the source momentum is simply 1/2 (uniform distribution). Thus the ensemble pmf can be r...