Momentum and momentum propensity
In an earlier post, we have seen that in the proposed digital QM model, evolution happens on nodes of a spatiotemporal lattice. At each iteration, position can increase of one lattice node in either directions or remain at the same it hold at the previous iteration. The corresponding possible values of momentum v are thus 1 (positive or "up" direction), 0 (rest), or -1 (negative or "down" direction).
In this post we will learn more on this random variable. Since v can only take three values, its probability distribution is completely defined by two parameters. These two parameters are, e.g., the expected value and the variance. The expected value of momentum plays a special role in the model and is called momentum propensity, here denoted as V (bold v in the papers). Instead of the momentum variance, we take the expected value of the momentum squared (a quantity that can take only the values 0 and 1), which we call energy propensity and denote as e.
The probability distribution of v is thus determined as
=\frac{e+V}{2},\quad&space;\Pr(v=0)=1-e,\quad&space;\Pr(v=-1)=\frac{e-V}{2} "\Pr(v=1)=\frac{e+V}{2},\quad \Pr(v=0)=1-e,\quad \Pr(v=-1)=\frac{e-V}{2}")
that resembles the special relativity's definition of energy (in lattice units).
The momentum propensity itself is a random variable, which can take rational values (see future posts on this point) between -1 and +1. Consequently, the energy propensity can take values from 1/2 to 1. In particular, we obtain that when V=1 or V=-1, also e=1. Consequently, the probability of staying at rest drops to zero and the particle always moves in direction "up" (if V=1) or "down" (if V=-1).
At a given iteration, the value of the momentum propensity results from the sum of two contributions, a quantum momentum, denoted as vQ, and an external force momentum, denoted as vF. In future posts, we shall see how these quantites are build up at the particle's preparation and change during its evolution.
In this post we will learn more on this random variable. Since v can only take three values, its probability distribution is completely defined by two parameters. These two parameters are, e.g., the expected value and the variance. The expected value of momentum plays a special role in the model and is called momentum propensity, here denoted as V (bold v in the papers). Instead of the momentum variance, we take the expected value of the momentum squared (a quantity that can take only the values 0 and 1), which we call energy propensity and denote as e.
The probability distribution of v is thus determined as
Note that the sum of the three probabilities is equal to one, as it should be. Of course, by subtracting the third probability from the first, one retrieves the expected value of v, that is, V. Similarly, by summing the first and the third probability, one retrieves the expected value of v2, that is, e, as it also should be.
Further, the model assumes that the energy propensity is related to the momentum propensity according to the equationThe momentum propensity itself is a random variable, which can take rational values (see future posts on this point) between -1 and +1. Consequently, the energy propensity can take values from 1/2 to 1. In particular, we obtain that when V=1 or V=-1, also e=1. Consequently, the probability of staying at rest drops to zero and the particle always moves in direction "up" (if V=1) or "down" (if V=-1).
At a given iteration, the value of the momentum propensity results from the sum of two contributions, a quantum momentum, denoted as vQ, and an external force momentum, denoted as vF. In future posts, we shall see how these quantites are build up at the particle's preparation and change during its evolution.
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